If $f(w) = g(\langle w, x \rangle + y )$ with $g$ a $\beta$-smooth function. Show $f$ is $\left(\beta \|x\|^2\right)$-smooth

Question

As the title says really.

I think I need to use a theorem that states:

For $\beta$-smooth $f$

$$f(v) \leq f(w) + \langle \nabla f(w), v-w \rangle + \frac{\beta}{2} \|v-w\| ^2.$$

Not sure how to prove it from this though.