What must be the value of the coefficient $c$ in $P(x)=x^3+2x^2+cx+10$, in order for $x-5$ to be a factor of $P$?
From the Factor Theorem, if $x-5$ is a factor of $P$, then $P(5) = 0$. Using this we have
$$5^3+2(5^2)+5c+10 = 0.$$Solving for $c$ gives us $c = \boxed{-37}$.