for g(x)=x^3 + 3x^2 -2, show that there must ne one real root on [0,2]
It's a useless detour (at least how I did it is), but since you want it so badly:
Let $\displaystyle g_c(x)= \frac{x^4}{4} + x^3 -2x +c=4x(x^3+4x^2-2)+c$ Let $\displaystyle h(x)=x^3+4x^2-2$ then $\displaystyle h(0)<0$ and $\displaystyle h(2)>0$ so by the intermediate value theorem there exists $\displaystyle d\in (0,2)$ such that $\displaystyle h(d)=0$ and so $\displaystyle g_c(0)=g_c(d)=c$ and by Rolle's theorem there exists $\displaystyle y\in(0,d)$ such that $\displaystyle g'_c (y)=0$ but $\displaystyle g'_c=g$
As skeeter said, not pretty at all.