Find the quadratic polynomial $p(x)$ such that $p(-7) = 0,$ $p(4) = 0,$ and $p(5) = -36.$
Since $p(-7) = p(4) = 0,$ the quadratic polynomial $p(x)$ is of the form
\[p(x) = c(x + 7)(x - 4),\]for some constant $c.$  To find $c,$ we set $x = 5,$ and use the fact that $p(5) = -36$:
\[-36 = c(12)(1),\]so $c = -3.$  Hence, $p(x) = -3(x + 7)(x - 4) = \boxed{-3x^2 - 9x + 84}.$