Calculating covariance, with multiplication by stochastic variable.

Calculating covariance, with multiplication by stochastic variable.

As an exercise I'm supposed to calculate; $\text{cov}(X \cdot Y,X)$, where
$X$ and $Y$ are independent discrete stochastic variables, with
probability functions given by; $$ p\left(var\right) = \left\{
\begin{array}{ll} 0.1 & \text{ if } var = 0 \\ 0.4 & \text{ if } var = 1
\\ 0.5 & \text{ if } var = 2 \\ 0 & \text{ otherwise.} \end{array} \right.
$$ However I'm unable to find a rule, which applies to this case. (That is
covariance, with multiplication by stochastic varibles).
I've tried to substitute $X \cdot Y$, with a new stochastic variable $Z =
X \cdot Y$, with $p_{Z}\left(z\right) = p_{X}\left(z\right) \cdot
p_{Y}\left(y\right)$, however this does not seem to result in the correct
answer.
So I'd love to get a pointer, on how to proceed.
The answer is supposed to be $0.616$, $EX = 1.4$, $Var\left(X\right) = 0.44$.