When finding the root of a number with an even exponent, $x^y$ becomes $\pm x$. How would this work in a situation such as $a = \sqrt{(5x + 12)^2 + m}$? I know that the result is not $a = \pm 5x + 12 + \sqrt m$ because the parentheses must be evaluated first.
Could anybody please tell me how to evaluate the answer to this problem correctly?
First off, the notation $\sqrt [n] x$, for positive $x$ and even $n$, refers specifically to the positive $n$th root of $x$. So evaluating $a$ as you've written it does not require any $\pm$s whatsoever. You are absolutely correct that you cannot just add together the square roots of the individual terms—in fact, the expression you've given is about as good as any; if you like you can expand the squared term, but that won't actually simplify anything.
If you're trying to solve the equation $a^2=(5x+12)^2+m$ for $a$, you will get $a=\pm\sqrt{(5x+12)^2+m}$.