Question 281598
There are two ways to completing this. I will show you the AC method.

Your problem states:

{{{4p^2 + 12pq + 9q^2}}}
We want to multiply the 4 and 9 together {{{ 4 * 9 = 36}}}

We want to multiply two things to get 36 and add up to be 12 (12 is the middle number and 36 is 4*9). The answer for this one would be 6 and 6 because:

{{{6 * 6 = 36}}}
and
{{{6 + 6 = 12}}} 

A couple steps more, divide the middle up into:

{{{4p^2 + 6pq + 6pq + 9q^2}}}

I divided up the 6pq and 6pq because those were our answers, lastly, factor by grouping:

{{{(4p^2 + 6pq) + (6pq + 9q^2)}}}
{{{2p(2p + 3q) + 3q(2p + 3q)}}} Key point in factor by grouping*
{{{(2p + 3q) (2p + 3q)}}}

To factor by grouping, you will see two of the same things, in our case it was {{{(2p + 3q)}}}, we take one of those and you will see {{{2p + 3q}}} remains

which is why we got
{{{(2p + 3q) (2p + 3q)}}}
{{{highlight((2p + 3q)^2)}}} final answer