Substitute f(x) = x^2 + a x + b into the integral equation

Substitute f(x) = x^2 + a x + b into the integral equation and compare the coefficients.

Then we obtain two linear equations in a and b:

a + b = - 2/3

3a + 2b = - 8/3

which yield a = - 4/3 and b = 2/3.

Answer:  f(x) = x^2 - (4/3) x + 2/3.