Polynomials through Integrals.

Let $a=\int^{1}_{-1}tf(t)\ dt$ and $b=\int^{1}_{-1}f(t)\ dt$ . Then, we have $f(x)=(a+1)x^2+bx^3.$ Substitute this into the original equation yields $\begin{aligned} f(x)&=&x^2+x^2\int^{1}_{-1}\left(t\left((a+1)t^2+bt^3\right)\right)\ dt+x^3\int^1_{-1}\left((a+1)t^2+bt^3\right)\ dt\\ &=& \left( \frac{2b}{5} +1 \right) x^2+\frac{2(a+1)}{3}x^3. \end{aligned}$ Comparing the coefficients of the like terms gives the following system of simultaneous equations. $\begin{cases} a&=\frac{2b}{5}\\ b&=\frac{2(a+1)}{3} \end{cases}$ Solving gives $a=\frac{4}{11}$ and $b=\frac{10}{11}$ and hence $f(x)=\frac{15}{11}x^2+\frac{10}{11}x^3.$ This gives $11f(1)=25.$