Answer
$x=-7$
Work Step by Step
We rewrite the 4 on the left, so that the base is $2$ on both sides.
$4^{x-2}=2^{3x+3}$
$(2^2)^{x-2}=2^{3x+3}$
By the law of exponents: $(a^x)^y=a^{xy}$
$(2^2)^{x-2}=2^{2x-4}$
Then we set the exponents equal, because the two sides with the same base are equal only if the powers are equal too: if $b^m=b^n$ and $b \ne0$ , $b \ne1$ then $m=n$
$2x-4=3x+3$
$-7=x$