Your attachment doesn't have anything to do with this question! Yes, a quadratic polynomial, P(x), such that P(0)= 1 and P'(0)= 0, is of the form P(x)= ax^2+ 1. The third condition is that "the integral of P(x) over x^3(x-1)^2 is a rational function".
That is . Do that integration by "partial fractions". We can write as .
The crucial point is that the integrals of and are logarithms, not rational functions. Can you find "a" such that A and D are 0?