Question

use the remainder theorem and the factor theorem to determine whether (c+5) is a factor of (c^4+7c^3+6c^2-18c+10)

Answer 1

When $c=-5$, we have

$(-5)^4+7(-5)^3+6(-5)^2-18(-5)+10=0$

which is the same as the remainder upon dividing $c^4+7c^3+6c^2-18c+10$ by $c+5$, so $c+5$ is indeed a factor of the quartic.