Question

Factorise the following :
$(p-q{)}^2-6(p-q)-16$

Answer 1

$(p-q{)}^2-6(p-q)-16$

$\Rightarrow p^2+q^2-2pq-6p+6q-16$

add and subtract $9$

$\Rightarrow p^2+q^2-2pq-6p+6q+9-16-9$

$\Rightarrow p^2+q^2+(3{)}^2+2(-p\left)\right(q)+2(3\left)\right(-p)+2(3\left)\right(q)-25$

$\Rightarrow (-p+q+3{)}^2-(5{)}^2$ $[\because a^2-b^2=(a+b\left)\right(a-b\left)\right]$

$\Rightarrow (q+3-p+5)(q+3-p-5)$

$\Rightarrow (q-p+8)(q-p-2)$.