When $Z \sim \mathcal{N}(\mu, \sigma^2)$ we have $M_{z}(t)= e^{\mu t + (\sigma t)^2/2}$. If we know $X \sim \mathcal{N}(\mu, \sigma^2)$ and $Y=4X+10$, show that $Y \sim \mathcal{N}(4 \mu + 10, 16 \sigma^2).$
i know that $$ M_Y(t) = \mathbb{E}\left[e^{tY}\right] = \mathbb{E}\left[e^{t(4X+10)}\right] = \mathbb{E}\left[e^{4tX} \cdot e^{10t}\right] $$
but how can i find the distributions?