I am not sure whether any of the previous answers really address the question, which is to have a trick to solve the equation $x^2+5x-1476=0$. Of course, the explicit formula for the roots gives the answer, but I would no qualify this as a trick, but as a generic solution for this type of problem.
There is however a trick, if you assume that, as most of the elementary questions of this type, the equation has integer solutions. In this case, you can write
$x^2+5x-1476= (x+a)(x-b)$ for some integers $a$ and $b$. It follows $ab = 1476$ and $a-b = 5$. Now comes the trick: the prime decomposition of $1476$ is $41 \times 2^2 \times 3^2$ (easy to find), which let you few possibilities for $a$. Just try the first one, $a = 41$, then $b = 2^2 \times 3^2 = 36$. Bingo! one gets $a-b = 5$, so we are done.