Find f: f''=20x^3 + 12x^2 + 4 f(0)=8 f(1)=5
$\displaystyle f''(x)=20x^3 + 12x^2 + 4$
$\displaystyle f'(x)=5x^4 + 4x^3 + 4x + C_1$
$\displaystyle f(x)=x^5 + x^4 + 2x^2 + C_1x + C_2$
now, substitute $\displaystyle x=0$ and $\displaystyle x=1$ and you will have 2 equations and two unknowns.
and then solve for $\displaystyle C_1$ and $\displaystyle C_2$.