How to factor this expression in the fireman's problem?

Question

I'm trying to solve the fireman's optimization problem. It boils down to factoring the following expression:

$2\left ( q + \frac{pq}{a} \right )\cdot \left ( -\frac{pq}{a^2} \right ) + 2(p+a)$

I have spent around 5 pages of paper trying to solve this but to no avail. According to the solution you can factor out $2(p+a)$, but how? This is not homework by the way.

Any help is appreciated.

Answer 1

Since $q+\frac{pq}{a}=\frac{q}{a}(a+p)$, your expression factorises to $$2(p+a)\left(\frac{q}{a}\frac{-pq}{a^2}+1\right)=2(p+a)(1-pq^2/a^3).$$

Answer 2

When dealing with fractions, it helps to clear denominators.

Let $$ t=2\left ( q + \frac{pq}{a} \right )\cdot \left ( -\frac{pq}{a^2} \right ) + 2(p+a) $$ Then $$ a^3t= 2\left ( aq + {pq} \right )\cdot \left ( -{pq} \right ) + 2(p+a)a^3 $$ Now just notice $$ (aq+pq)=(a+p)q=(p+a)q $$