Co-variance of two dependent Gaussian variables

Question

I have a Gaussian variable $x\sim N(\mu,\sigma_1^2)$. Also $y=cx+v$, where $c$ is just soem constant and $v\sim N(0,\sigma_2^2)$.

So one could see that $y$ is also a gaussian variable, $y\sim N(c\mu, c^2\sigma_1^2+\sigma_2^2)$.

Now I have to calculate the covariance of $x$ and $y$.

$cov(x,y)=E[(x-\mu)(y-c\mu)]=E[xy]-c\mu^2$

Now calculating E[xy] is tricky because x and y are not independent. What trick should be used in this case?

Answer 1

$$E[xy]=E[x(cx+v)]=E[cx^2+xv]$$ and $x$ and $v$ are independent.