MathNet 2007-part-02 — 187 problems (EN–VI) Question 152

Let $a$, $b$, and $c$ be real constants such that $x^{2}+x+2$ is a factor of $a x^{3}+b x^{2}+c x+5$, and $2 x-1$ is a factor of $a x^{3}+b x^{2}+c x-\frac{25}{16}$. Find $a+b+c$.