a) a³ - 7a - 6
= a³ + a² - a² - a - 6a - 6
= a²(a + 1) - a(a + 1) - 6(a + 1)
= (a + 1)(a² - a - 6)
= (a + 1)(a² - 3a + 2a - 6)
= (a + 1)[a(a - 3) + 2(a - 3)]
= (a + 1)(a - 3)(a + 2)
b) a³ + 4a² - 7a - 10
= a³ - 2a² + 6a² - 12a + 5a - 10
= a²(a - 2) + 6a(a - 2) + 5(a - 2)
= (a - 2)(a² + 6a + 5)
= (a - 2)(a² + a + 5a + 5)
= (a - 2)[a(a + 1) + 5(a + 1)]
= (a - 2)(a + 1)(a + 5)
c) a(b + c)² + b(c + a )² + c(a + b )² - 4abc
= ab² + bc² + 2abc + bc² + ba² + 2abc + ca² + cb² + ...
= ab² + ac² + bc² + ba² + ca² + cb² + 2abc
= ab(b + c) + ac(b + c) + cb(b + c) + a²(b + c)
= (b + c)[a(a + c) + b(a + c)]
= (a + b)(a + c)(b + c)
e) (x² + x + 1)( x² + x + 2) - 12
= (x² + x + 1)[(x² + x + 1) + 1] - 12
= (x² + x + 1)² + (x² + x + 1) - 12
= (x² + x + 1)² - 3(x² + x + 1) + 4(x² + x + 1) - 12
= (x² + x + 1)(x² + x + 1 - 3) + 4(x² + x + 1 - 3)
= (x² + x + 1)(x² + x - 2) + 4(x² + x - 2)
= (x² + x + 2)(x² + x + 1 + 4)
= (x² - x + 2x - 2)(x² + x + 5)
= [x(x - 1) + 2(x - 1)](x² + x + 5)
= (x - 1)(x + 2)(x² + x + 5)
g) x⁸ + x + 1
= $x^{8}$ + $x^{7}$ - $x^{7}$ + $x^{6}$ - $x^{6}$ + $x^{5}$ - $x^{5}$ + $x^{4}$ - $x^{4}$ + $x^{3}$ - $x^{3}$ + $x^{2}$ - $x^{2}$ + x + 1
= ($x^{8}$ + $x^{7}$ + $x^{6}$) - ($x^{7}$ + $x^{6}$ + $x^{5}$) - ($x^{5}$ + $x^{4}$ + $x^{3}$) - ($x^{4}$ + $x^{3}$ + $x^{2}$) + ($x^{2}$ + x + 1)
= $x^{6}$($x^{2}$ + x + 1) - $x^{5}$($x^{2}$ + x + 1) + $x^{3}$($x^{2}$ + x + 1) - $x^{2}$($x^{2}$ + x + 1) + ($x^{2}$ + x + 1)
= ($x^{2}$ + x + 1)($x^{6}$ - $x^{5}$ + $x^{3}$ - $x^{2}$ + 1)
