f(x)=2x^2+2mx+12
If f(x-3) is even, what is the value of "m"?
That is not true. A function is even if $\displaystyle \displaystyle \begin{align*} f(-x) = f(x) \end{align*}$ for all x.
Consider the function $\displaystyle \displaystyle \begin{align*} f(x) = (x - 1)^2 \end{align*}$, then $\displaystyle \displaystyle \begin{align*} f(-x) = (-x-1)^2 \neq (x-1)^2 \end{align*}$.
The OP needs to check what f(x-3) is and then set it equal to f( -(x-3) ).