∵ ax² + (a + b)x + b

= ax² + ax + bx + b

= ax(x + 1) + b(x + 1)

= (x + 1)(ax + b)

∵ 3x² + 14x + 8

= 3x² + (12 + 2)x + 8

= 3x² + 12x + 2x + 8

= 3x(x + 4) + 2(x + 4)

= (x + 4)(3x + 2)

∵ 6x² - x - 15

= 6x² - (10 - 9)x - 15

= 6x² - 10x + 9x - 15

= 2x(3x - 5) + 3(3x - 5)

= (3x - 5)(2x + 3)

∵ 9r² + r - 8

= 9r² + (9 - 8)r - 8

= 9r² + 9r - 8r - 8

= 9r(r + 1) - 8(r + 1)

= (r + 1)(9r- 8)

∵ 6m² - 11mn - 10n²

= 6m² - (15 - 4)mn - 10n²

= 6m² - 15mn + 4mn - 10n²

= 3m(2m - 5n) + 2n(2m - 5n)

= (2m - 5n)(3m + 2n)

∵ 7p² + 48pq - 7q²

= 7p² + (49 - 1)pq - 7q²

= 7p² + 49pq - pq - 7q²

= 7p(p + 7q) - q(p + 7q)

= (p + 7q)(7p - q)

∵ 12 + a - 6a²

= 12 + (12 - 6)a - 6a²

= 12 + 12a - 6a - 6a²

= 12(1 + a) - 6a(1 + a)

= (1 + a)(12 - 6a)

= (1 + a)6(2 - a)

= 6(1 + a)(2 - a)

∵ 6 + 5a - 6a²

= 6 + (9 - 4)a - 6a²

= 6 + 9a - 4a - 6a²

= 3(2 + 3a) - 2a(2 + 3a)

= (2 + 3a)(3 - 2a)

∵ 99a² - 202ab + 99b²

= 99a² - (121 + 81)ab + 99b²

= 99a² - 121ab - 81ab + 99b²

= 11a(9a - 11b) - 9b(9a - 11b)

= (9a - 11b)(11a - 9b)

∵ ax² + (a² + 1)x + a

= ax² + a²x + x + a

= ax(x + a) + (x + a)

= (x + a)(ax + 1)

∵

=

=

=

=

∵ x² + (a + b)x + ab

= x² + ax + bx + ab

= x(x + a) + b(x + a)

= (x + a)(x + b)

∵ x² + 7x + 12

= x² + (4 + 3)x + 4 × 3

= x² + 4x + 3x + 4 × 3

= x(x + 4) + 3(x + 4)

= (x + 4)(x + 3)

∵ x² + 7x - 18

= x² + (9 - 2)x + 9 × (-2)

= x² + 9x - 2x - 9 × 2

= x(x + 9) - 2(x + 9)

= (x + 9)(x - 2)

∵ x² + 13x - 48

= x² + (16 - 3)x - 16 × 3

= x² + 16x - 3x - 16 × 3

= x(x + 16) - 3(x + 16)

= (x + 16)(x - 3)

∵ (a + b)² - 5(a + b) - 6

ধরি, (a + b) = m

∴ m² - 5m - 6

= m² - (6 - 1)m - 6

= m² - 6m + m - 6

= m(m - 6) + (m - 6)

= (m - 6)(m + 1)

= (a + b - 6)(a + b + 1)

∵ (p² - 3q²)² - 16(p² - 3q²) + 63

ধরি, (p² - 3q²) = m

∴ m² - 16m + 63

= m² - (7 + 9)m + 63

= m² - 7m - 9m + 63

= m(m - 7) - 9(m - 7)

= (m - 7)(m - 9)

= (p² - 3q² - 7)(p² - 3q² - 9)

∵ a² + ab - 12b²

= a² + (4 - 3)ab - 12b²

= a² + 4ab - 3ab - 12b²

= a(a + 4b) - 3b(a + 4b)

= (a + 4b)(a - 3b)

∵ (a + b)² - 5a - 5b + 6

= (a + b)² - 5(a + b) + 6

ধরি, (a + b) = m

∴ m² - 5m + 6

= m² - (3 + 2)m + 6

= m² - 3m - 2m + 6

= m(m - 3) - 2(m - 3)

= (m - 3)(m - 2)

= (a + b - 3)(a + b - 2)

∵ x⁴ - 7x² + 12

= (x²)² - 7x² + 12

ধরি, x² = m

∴ m² - 7m + 12

= m² - (4 + 3)m + 12

= m² - 4m - 3m + 12

= m(m - 4) - 3(m - 4)

= (m - 4)(m - 3)

= (x² - 4)(x² - 3)

= (x² - 2²)(x² - 3)

= (x - 2)(x + 2)(x² - 3)

∵ x⁶y⁶ - 9x³y³ + 8

= (x³y³)² - 9x³y³ + 8

ধরি, x³y³ = m

= m² - 9m + 8

= m² - (8 + 1)m + 8

= m² - 8m - m + 8

= m(m - 8) - (m - 8)

= (m - 8)(m - 1)

= (x³y³ - 8)(x³y³ - 1)

= {(xy)³ - 2³}{(xy)³ - 1³}

= {(xy) - 2}{(xy)² + (xy).2 + 2²}{(xy) - 1}{(xy)² + (xy) + 1²}

= (xy - 2)(x²y² + 2xy + 4)(xy - 1)(x²y² + xy + 1)

∵ x² − 2ax + (a + b)(a − b)

= x² − {(a + b) + (a − b)} x + (a + b)(a − b)

= x² − (a+b)x − (a − b)x + (a + b)(a − b)

= x{x − (a + b)} − (a − b){x − (a + b)}

= {x − (a + b)}{x − (a − b)}

= (x − a − b)(x − a + b)

∵ x² − bx - (a + 3b)(a + 2b)

= x² − {(a + 3b) - (a + 2b)} x - (a + 3b)(a + 2b)

= x² − (a + 3b)x + (a + 2b) x - (a + 3b)(a + 2b)

= x{x − (a + 3b)} + (a + 2b){x − (a + 3b)}

= {x − (a + 3b)}{x + (a + 2b)}

= (x − a − 3b)(x + a + 2b)

∵ p² + p - (a + 1)(a + 2)

= p² + {(a + 2) - (a + 1)} p - (a + 1)(a + 2)

= p² + (a + 2)p - (a + 1) p - (a + 1)(a + 2)

= p{p + (a + 2)} - (a + 1){p + (a + 2)}

= {p + (a + 2)}{p - (a + 1)}

= (p + a + 2)(p - a - 1)

∵ (x² + 1)² - (x² - 1) - 4x²

= (x² - 1)² + 4.x².1 - (x² - 1) - 4x²

= (x² - 1)² + 4x² - (x² - 1) - 4x²

= (x² - 1)² - (x² - 1)

= (x² - 1){(x² - 1) - 1}

= (x² - 1²)(x² - 2)

= (x + 1)(x - 1)(x² - 2)

∵ x² + 2(a² + b²)x + (a² - b²)²

= x² + 2(a² + b²)x + {(a + b)(a - b)}²

= x² + {(a + b)² + (a - b)²}x + (a + b)²(a - b)²

= x² + (a + b)²x + (a - b)²x + (a + b)²(a - b)²

= {x + (a + b)²}x + (a - b)²{x + (a + b)²}

= {x + (a + b)²}{x + (a - b)²}

= (x + a² + 2ab + b²)(x + a² - 2ab + b²)