Expression for Sum of Multivariate Gaussian Random Vectors

Question

Consider two multidimensional random vectors $x$ and $z$ having Gaussian distributions $P(x)=N(x\mid\mu_x,\Sigma_x)$ and $P(z)=N(z\mid\mu_z,\Sigma_z)$, respectively, together with their sum $y=x+z$. Find an expression for the marginal distribution $P(y)$ by considering the linear-Gaussian model comprising the product of the marginal distribution $P(x)$ and the conditional distribution $P(y\mid x)$.

Answer 1

Is it me, or can one forget all the specifics of the question and simply rely on the well-known Bayes formula $f_Y(y)=\displaystyle\int f_{Y\mid X}(y\mid x)f_X(x)\mathrm dx$?