Find variance of $-3X+5$ given $E(X)$ and $E(X(X-2))$

Question

Suppose that $X$ is a discrete random variable with $E(X) = 1$ and $E[X(X −2)]= 3$. Find $Var(−3X + 5)$.

$$E[X(X-2)]=E(X^2-2X)=3$$ $$Var(-3X+5)=E[(-3X+5-1)^2 ]=E(9X^2-24X+16)$$ $$E(9X^2-24X+16)=9E(X^2-2X)-6E(X)+16=37$$

I found the answer to be $27-6+16=37$, but the solution manual says the answer is 36. I've triple-checked the last calculation and don't believe I've made an error, so I'm not sure why my answer is wrong.

Answer 1

$var (-3X+5)=E(-3X+5-2)^{2}$ since $E(-3X+5)=-3+5=2$. So your second line is wrong. The correct answer is $36$.