Hexagons!

Using right $\triangle{ADG}$ in the diagram above we obtain:

$|\overline {\rm AD}|^2 = (12 + x)^2 + 3^2 = 153 + 24x + x^2 = (15\sqrt{2})^2 = 450$

$\implies x^2 + 24x - 297 = 0 \implies (x + 33)(x - 9) = 0 \implies x = -33$ or $x = 9$ .

Since $x = -33$ is not a valid solution to this problem $\implies x \neq -33 \implies \boxed{x = 9}$ .