Institution: Bangladesh Public Service Commission (BPSC বিসিএস ) Exam: 32th BCS Preliminary Exam 2012
Topic: Arithmetic Mathematics (পাটিগণিত), Subject: Mathematics (সাধারণ গণিত)

Best Answer Explained in Below-:

<p data-sourcepos="1:1-1:52" style="margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">x^2 - y^2 + 2y - 1 এর একটি উৎপাদক হলো <span style="font-weight: 700;">x + y - 1</span>।</p><p data-sourcepos="3:1-3:13" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;"><span style="font-weight: 700;">ব্যাখ্যা:</span></p><ol data-sourcepos="5:1-6:0" style="margin-top: 4px; margin-bottom: 4px; padding-inline-start: 1.75rem; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif;"><li data-sourcepos="5:1-6:0"><span style="font-weight: 700;">বর্গ পূরণ:</span></li></ol><p data-sourcepos="7:1-7:55" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">প্রথমে, আমরা x^2 - y^2 + 2y - 1 কে একটি বর্গ পূর্ণ করব।</p><p data-sourcepos="9:1-9:24" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">y^2 - 2y + 1 = (y - 1)^2</p><p data-sourcepos="11:1-11:46" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">এখন, সমীকরণটিতে (y - 1)^2 যোগ এবং বিয়োগ করবো:</p><p data-sourcepos="13:1-13:42" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">x^2 - y^2 + 2y - 1 + (y - 1)^2 - (y - 1)^2</p><p data-sourcepos="15:1-15:33" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">= x^2 - (y - 1)^2 + (y - 1)^2 - 1</p><p data-sourcepos="17:1-17:28" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">= (x + y - 1)(x - y + 1) - 1</p><ol data-sourcepos="19:1-20:0" start="2" style="margin-top: 4px; margin-bottom: 4px; padding-inline-start: 1.75rem; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif;"><li data-sourcepos="19:1-20:0"><span style="font-weight: 700;">উৎপাদক বের করা:</span></li></ol><p data-sourcepos="21:1-21:85" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">উপরের সমীকরণ থেকে দেখা যাচ্ছে, x^2 - y^2 + 2y - 1 এর একটি উৎপাদক হলো <span style="font-weight: 700;">(x + y - 1)</span>।</p><p data-sourcepos="23:1-23:32" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;"><span style="font-weight: 700;">অন্যান্য বিকল্প পরীক্ষা করা:</span></p><p data-sourcepos="25:1-25:54" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">অন্যান্য বিকল্পগুলি x^2 - y^2 + 2y - 1 এর উৎপাদক নয়।</p><ul data-sourcepos="27:1-28:0" style="margin-top: 4px; margin-bottom: 4px; padding-inline-start: 1.125rem; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif;"><li data-sourcepos="27:1-28:0"><span style="font-weight: 700;">x + y + 1:</span></li></ul><p data-sourcepos="29:1-29:58" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">(x + y + 1)(x - y + 1) = x^2 + y^2 + 1 + x - y + x - y + 1</p><p data-sourcepos="31:1-31:20" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">≠ x^2 - y^2 + 2y - 1</p><ul data-sourcepos="33:1-34:0" style="margin-top: 4px; margin-bottom: 4px; padding-inline-start: 1.125rem; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif;"><li data-sourcepos="33:1-34:0"><span style="font-weight: 700;">x - y:</span></li></ul><p data-sourcepos="35:1-35:36" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">(x - y)(x + y) = x^2 + xy - xy - y^2</p><p data-sourcepos="37:1-37:11" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">= x^2 - y^2</p><p data-sourcepos="39:1-39:20" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">≠ x^2 - y^2 + 2y - 1</p><ul data-sourcepos="41:1-42:0" style="margin-top: 4px; margin-bottom: 4px; padding-inline-start: 1.125rem; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif;"><li data-sourcepos="41:1-42:0"><span style="font-weight: 700;">x - y - 1:</span></li></ul><p data-sourcepos="43:1-43:56" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">(x - y - 1)(x + y - 1) = x^2 + xy - x - y - xy - y^2 + 1</p><p data-sourcepos="45:1-45:15" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">= x^2 - y^2 - 1</p><p data-sourcepos="47:1-47:20" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">≠ x^2 - y^2 + 2y - 1</p><p data-sourcepos="49:1-49:10" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;"><span style="font-weight: 700;">উত্তর:</span></p><p data-sourcepos="51:1-51:52" style="margin-top: 1rem; margin-right: 0px; margin-left: 0px; white-space-collapse: preserve; word-break: break-word; color: rgb(31, 31, 31); font-family: &quot;Google Sans&quot;, &quot;Helvetica Neue&quot;, sans-serif; font-size: 16px;">x^2 - y^2 + 2y - 1 এর একটি উৎপাদক হলো <span style="font-weight: 700;">x + y - 1</span>।</p>