Addition, Multiplication, and Substitution of Taylor Series

Let $P$ be the quadratic function such that $P(0) = 7$, $P(1) = 10$, and $P(2) = 25$. If $a$, $b$, and $c$ are integers such that every positive number $x$ less than 1 satisfies $$\sum_{n = 0}^\infty P(n) x^n = \frac{ax^2 + bx + c}{{(1 - x)}^3},$$ compute the sum $a+b+c$.

Source: Math Prize for Girls