Suppose X and Z are independent random variables, where X is standard normal and p(Z = -1) = p(Z =1) = 1/2. Let Y = XZ be the product. Show that Y follows N(0,1).

Note that, I have the solution start from P(Y x) = P(Z = -1, -X x) + P(Z = 1, X x).

I don't know where does this equation come from, especially why there is -X x. What is the general formula to solve the product of two distributions?