For non-zero integers $x$ and $y$ we define $x \nabla y=x^{y}+4 x$. For example, $2 \nabla 3=2^{3}+4(2)=16$. Determine all real numbers $x$ such that $x \nabla 2=12$.
We assume $12=x \nabla 2=x^{2}+4 x$, so $x^{2}+4 x=12$.

This can be rearranged to $x^{2}+4 x-12=0$, which factors as $(x+6)(x-2)=0$.

The possible values of $x$ are $x=2$ and $x=-6$.

ANSWER: $-6,2$