TITLE: Stuck proving that if $m$ and $n$ are perfect squares. Then $m+n+2\sqrt{mn}$ is also a perfect square.
QUESTION [5 upvotes]: I am relatively new to proofs and can't seem to figure out how to solve an exercise.
I am trying to prove:

Suppose that $m$ and $n$ are perfect squares.  Then $m+n+2\sqrt{mn}$ is also a perfect square.

I know that per the definition of a perfect square, that $m=a^2$ and $n=b^2$, if a and b are some positive integer.
I can then use substitution to rewrite the statement as:
$$a^2+b^2+2\sqrt{a^2b^2}$$
I also know that $2\sqrt{a^2b^2}$ can be simplified to:
S
$$a^2+b^2+2ab$$
I am stuck after this point though.  I don't know how to eliminate the $2ab$.

REPLY [4 votes]: You don't need to eliminate the $2ab$ term. 
Notice that $(a+b)^2 = (a+b)(a+b) = a^2+ab+ba+b^2 = a^2+b^2+2ab$.

REPLY [2 votes]: Now use the fact that $a^2+b^2+2ab = (a+b)^2$.