We use the well known inequality $a^2 + b^2 \ge 2ab$ to adjust the given one:
$$4a = (a + c)(b^2 + ac) = a(b^2 + c^2) + c(a^2 + b^2) \ge a(b^2 + c^2) + 2abc = a(b + c)^2.$$
We can see, that $b + c \le 2$, and also that the equality holds if and only if $0 < a = b < 2$ and $c = 2 - b > 0$. That's it.