One way is to consider a line b = k * a for some number k and to prove that the inequality holds for all points (a, b) on this line. (For completeness, one should consider the line a = 0, but the inequality is symmetric w.r.t. a and b, so this is similar to k = 0.)
The substitution gives rise a quadratic inequality on where coefficients are quadratic polynomials of . By considering the leading coefficient and the discriminant, one can show that the inequality holds.