Let $X \sim N(-1,4)$, express the following probability $$P(X^2 + X \leq 0)$$ in terms of the cdf of the standard normal distribution $\Phi$.
I am trying this exercise now but as far as I know the multiplication of two gaussian is NOT gaussian. So what would the best method be to start here?
Edit regarding lulu‘s comment: if $X^2 + X \leq 0$, then by factorization $X \leq 0 \land X + 1 \geq 0$ So $ -1 \leq X \leq 0$
You just need to find for what $X$, $X^2+X\leq 0$ and then proceed: $$\mathcal{P}(X^2+X\leq 0)=\mathcal{P}(-1\leq X\leq 0)= \dfrac{1}{2\sqrt{2\pi}}\int_{-1}^0\exp\left(-\dfrac{1}{2}\left(\dfrac{x+1}{2}\right)^2\right)dx\approx 0.19.$$
I took into account that $N(\mu=-1,\sigma^2=4)=\dfrac{1}{\sigma\sqrt{2\pi}}\exp\left(-\dfrac{1}{2}\left(\dfrac{x-\mu}{\sigma}\right)^2\right)=\dfrac{1}{2\sqrt{2\pi}}\exp\left(-\dfrac{1}{2}\left(\dfrac{x+1}{2}\right)^2\right)$.