Let $A \sim Normal(k, 1)$ (ie. with mean $k$ and variance $1$). Let $B = A^2$. The first step is to express the CDF of $B$ in terms of the CDF $S_X(x)$ of the standard normal distribution (with mean $0$ and variance $1$).
That part I think is fine. I said the CDF of $B$ is expressed by: $P(B \leq b) = P(A^2 \leq b) = P(-\sqrt{b} \leq A \leq \sqrt{b}) = S_X(\sqrt{b} - k) - S_X(-\sqrt{b} - k)$.
The next part is to find the expectation and variance of $B$. I am not seeing how to do that short of evaluating some unappealing integrals. How would one go about doing this?
My class has not yet introduced the Chi-squared distribution, though I am aware this question is related to that. If possible, answers not using the Chi-squared distribution would be more helpful.
Edit: haven't covered moment-generating functions yet, either.
Note that $E(B) = E((Z+k)^2)$, where $Z$ ~ $N(0, 1)$. Do you know $E(Z^4)$?