If $Y| \mu \sim N(\mu, A)$ and $\mu \sim N(\mu_0, B)$, showing the marginal of $Y$ is $Y \sim N(\mu_0, A+B)$

Question

I have an n-dimensional multivariate distribution:

$$ Y| \mu \sim N(\mu, A) $$

and a prior of:

$$ \mu \sim N(\mu_0, B) $$

I would like to show that the marginal distribution of $Y$ is:

$$ Y \sim N(\mu_0, A+B) $$

My approach is to recognize:

$$ p(Y) = \int_{-\infty}^{\infty} p(Y|\mu) p(\mu) d\mu $$

This ultimately yields:

$$ p(Y) = \int_{-\infty}^{\infty} 2^{-\frac{k}{2}}|A|^{-\frac{1}{2}} e^{-\frac{1}{2}(y-\mu)^{T}A^{-1}(y-\mu)} 2^{-\frac{k}{2}}|B|^{-\frac{1}{2}} e^{-\frac{1}{2}(\mu-\mu_0)^{T}B^{-1}(\mu-\mu_0)} d\mu $$

To solve this is a mess and I am stuck here. Is there a simpler way to find the marginal? Thanks.