If $f(z)$ is entire and $|f''(z)| \leq 5$. Show that $f$ must have the form of $az^2 +bz +c$

Question

This might be a completely incorrect approach, but since we have that $|f''(z)| \leq 5$, can we integrate both sides twice and obtain that $|f(z)| \leq \frac{5z^2}{2} +cz +c$ for c being some constant?. Then by taking $|z| = R$ we would have that $|f(z)| \leq \frac{5R^2}{2} +cR +c$ and then move into applying cauchys estimate here, or is that a wrong way to approach this? I have a feeling I can't just integrate right away, or that integrating immediately is the wrong approach for some reason.