You can follow Henry's comments to arrive at the answer. However, another way to come to the answer is to use the fact that if $X$ and $Y$ are independent, then $Y | X = Y$ and $X |Y = X$.
By iterated expectations and variance expressions
\begin{align*} \text{Var}(XY) & = \text{Var}[\,\text{E}(XY|X)\,] + \text{E}[\,\text{Var}(XY|X) \,]\\ & = \text{Var}[\,X\, \text{E}(Y|X)\,] + E[\,X^2\, \text{Var}(Y|X)\,]\\ & = \text{Var}[\,X\, \text{E}(Y)\,] + E[\,X^2\, \text{Var}(Y)\,]\\ & = E(Y)^2\, \text{Var}(X) + \text{Var}(Y) E(X^2)\,. \end{align*}