Question:
Factorize:
$2(x+y)^{2}-9(x+y)-5$
Solution:
We have:
$2(x+y)^{2}-9(x+y)-5$
Let :
$(x+y)=u$
Thus, the given expression becomes
$2 u^{2}-9 u-5$
$2 u^{2}-9 u-5$
Now, we have to split $(-9)$ into two numbers such that their sum is $(-9)$ and their product is $(-10)$.
Clearly, $-10+1=-9$ and $-10 \times 1=-10$
$\therefore 2 u^{2}-9 u-5=2 u^{2}-10 u+u-5$
$=2 u(u-5)+1(u-5)$
$=(u-5)(2 u+1)$
Putting $u=(x+y)$, we get:
$2(x+y)^{2}-9(x+y)-5=(x+y-5)[2(x+y)+1]$
$=(x+y-5)(2 x+2 y+1)$
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