Question

Factorize:
2(x + y)² − 9(x + y) − 5

Answer 1

We have:
$$\begin{gathered} 2{\left(x+y\right)}^2-9\left(x+y\right)-5 \\ \mathrm{Let}: \\ (x+y)=u \end{gathered}$$
Thus, the given expression becomes
$2u^2-9u-5$
Now, we have to split ($-$9) into two numbers such that their sum is ($-$9) and their product is ($-$10).
Clearly, $-10+1=-9\mathrm{and}-10\times 1=-10$.

$$\begin{gathered} \therefore 2u^2-9u-5=2u^2-10u+u-5 \\ =2u(u-5)+1(u-5) \\ =(u-5)(2u+1) \end{gathered}$$
Putting $u=(x+y)$, we get:
$$\begin{gathered} 2{\left(x+y\right)}^2-9\left(x+y\right)-5=\left(x+y-5\right)\left[2\left(x+y\right)+1\right] \\ =\left(x+y-5\right)\left(2x+2y+1\right) \end{gathered}$$