We show that \[ \left|P(a) + P(b)\right| < c \implies |a + b| < k \]WLOG we can have $P$ be monic and depressed. WLOG only consider when $a > C > -C > b, |P(a)| > C, |P(b)| > C$ for some large $C > 0$ such that $P'(x)$ is fixed sign on $(C, \infty)$ and $(-\infty, -C)$, taking $k > 2C$. If $P$ has even degree, then taking $C > \frac{c}{2}$ gives the result. Else, we show that $P(a) \in \left(-P(-a-k), -P(-a+k)\right)$. Except, we have that asymptotically, $-P(-a-k) > P(a) > -P(-a+k)$ which finishes when updating $C$ for this hold to as well.