Maximum and Minimum Roots

The maximum and minimum values can be attained by changing the value of c such that one of the turning points on the graph is on the x-axis. That is to say, the graph is tangent to the x-axis and two of the roots are equal.

So, let $x, x$ and $y$ be the roots of the equation for local minima and maxima. (Maxima and minima are attained when two of the roots are equal. By vieta's formulas,

$2x+y=2$

Then, by Newton's Sums,

$2x^2+y^2=12$

Solving the systems gives us $y = (\frac{10}{3}, -2)$ So, the minimum is $-2$ and the maximum is $\frac{10}{3}$

So, $\frac{10}{3}-(-2) = \frac{16}{3}$

So, our answer is $16+3 = \boxed{19}$

Rearranging the equation $-c=x^3-2x^2-4x$ ,now we can easily plot RHS

As $c$ is real, It can easily be seen from graph,that the maximum value of $x$ for $3$ solutions $\text{{Not necessarily distinct}}$ is $\dfrac{10}{3}$ and minimum value is $-2$

Therefore $\Large{\dfrac{10}{3}+2=\color{#D61F06}{\boxed{\color{skyblue}{\boxed{\color{#EC7300}{\dfrac{16}{9}}}}}}}$