` 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)`