Question

Factorise:
x² − y² − 2y − 1

Answer 1

We have:
$x^2-y^2-2y-1=x^2-\left(y^2+2y+1\right)$
$$\begin{gathered} ={\left(x\right)}^2-{\left(y+1\right)}^2 \\ =\left\{x+\left(y+1\right)\right\}\left\{x-\left(y+1\right)\right\} \\ =\left(x+y+1\right)\left(x-y-1\right) \end{gathered}$$

∴ $x^2-y^2-2y-1=\left(x+y+1\right)\left(x-y-1\right)$