Say I have $X \sim N(\mu, \sigma^2)$. I would like to show
$$ Var(a_1X^2 + a_2X) > 0 $$
only except when $a_1 = a_2 = 0$
My attempt
$$ Var(a_1X^2 + a_2X) = Var(a_1 (X+a_2/(2a_1))^2 $$
where $X+a_2/(2a_1)$ follows scaled noncentral $\chi^2$. But I can't proceed.