Prove the Hanson–Wright inequality

Let $X = (X_1,\dots,X_n)$ have independent, mean-zero, sub-Gaussian coordinates with $\|X_i\|_{\psi_2} \le K$, and let $A \in \R^{n\times n}$. Prove that for every $t \ge 0$,

$$\P\Big(\big|X^\top A X - \E X^\top A X\big| \ge t\Big) \ \le\ 2\exp\left[-c\min\left(\frac{t^2}{K^4\|A\|_F^2},\ \frac{t}{K^2\|A\|}\right)\right],$$

with $c$ an absolute constant. Then explain why both norms of $A$ must appear.