jmc

Though it is possible to solve this problem using polynomial long division, it is quicker to use the Factor Theorem.

Let $f(x)=c{x}^3+19x^2-3cx+35$. If $x+7$ is a factor of $f(x)$, the factor theorem tells us that $f(-7)=0.$ Then $$c(-7{)}^3+19(-7{)}^2-3c(-7)+35=0,$$which simplifies to $-322c+966=0.$ We can solve for $c$ to get $c=\boxed{3}$.