Real roots and variation of the constant term for a cubic polynomial

Calculus Level 4

Let $P(x)=x^3+x^2+c$ a polynomial where $c$ is a real number. Then there is a finite interval $I$ such that, $P(x)$ has more than one real root if and only if $c$ is in $I$ . If the length of $I$ can be represented as a number of the form $\frac{a}{b},$ where $a$ and $b$ are coprimes, then find $a+b.$

Real roots and variation of the constant term for a cubic polynomial

If you liked this problem you can try a related problem.