Let $f(x)$ denotes the quadratic expression $\quad f(x) \equiv 3 a x^2+2 b x+c, \quad$ whose antiderivative be denoted by $f(x)=a x^3+b x^2+c x$
Now $f(x)$ being a polynomial in $R, f(x)$ is continuous and differentiable on $R$. To apply Rolle's theorem.
We observe that $f(0)=0$ and $f(1)=a+b+c=0$, by hypothesis. So there must exist at least one value of $x$, say $x=\alpha \in(0,1)$ such that
$f^{\prime}(\alpha)=0 \Leftrightarrow 3 a \alpha^2+2 b \alpha+c=0$
That is,
$f^{\prime}(x)=3 a x^2+2 b x+c=0$ has at least one root in [o, 1].