\begin{align*} 3(y + 1)^2 - 5y - 7 & = 3(y + 1)^2 - 5y - 5 - 2 \\ & = 3(y + 1)^2 - 5(y + 1) - 2 \\ \\ \text{From (a), } 3x^2 - 5x - 2 & = (3x + 1)(x - 2) \\ \\ \text{Replace } & x \text{ by } y + 1, \\ 3(y + 1)^2 - 5(y + 1) - 2 & = [3(y + 1) + 1](y + 1 - 2) \\ & = (3y + 3 + 1)(y - 1) \\ & = (3y + 4)(y - 1) \end{align*}

\begin{align*} x^2 - y^2 & = (x + y)(x - y) \qquad [ a^2 - b^2 = (a + b)(a - b)] \end{align*}

\begin{align*} x^2 - y^2 & = (x + y)(x - y) \\ \\ z = 70^2 - 3^2 & = (70 + 3)(70 - 3) \\ & = (73)(67) \\ & = 4891 \\ \\ \text{Factors: } & 1, 67, 73, 4891 \end{align*}

\begin{align*} 2x^4 - 2x^3 - 12x^2 & = 2x^2 (x^2 - x - 6) \qquad [\text{Factorise common factor, then quadratic}] \\ & = 2x^2 (x - 3)(x + 2) \end{align*}

\begin{align*} 6 + 3x - 24x^2 - 12x^3 & = 3(2 + x - 8x^2 - 4x^3) \qquad \qquad [\text{Factorise common factor}] \\ & = 3[ (2 + x) - 4x^2 (2 + x)] \qquad \quad [\text{Grouping}] \\ & = 3[ (1)(2 + x) - 4x^2 (2 + x)] \\ & = 3(2 + x)(1 - 4x^2) \\ & = 3(2 + x)[(1)^2 - (2x)^2] \\ & = 3(2 + x)(1 + 2x)(1 - 2x) \qquad \quad[ a^2 - b^2 = (a + b)(a - b)] \end{align*}