Normal distribution - Marginalization $P(x)$ given $P(z)$ and $P(x\mid z)$

Question

I'm trying to get $P(x)$ given $P(z)$ and $P(x\mid z)$. I tried doing P(x) = P(x|z)P(z) but I didn't manage to arrive at the correct result.

$P(z) = N(z\mid 0, I)$

$P(x\mid z) = N(x\mid Wz+ \mu, \phi)$

And the given solution for $P(x)$ is $N(\mu, WW^T + \phi)$.

Is it correct assuming $P(x) = P(x) = P(x\mid z)P(z)$? Otherwise, how would we get $P(x)$?

Thank you in advance.