Real roots and coefficients for a fourth degree polynomial

Both roots of the polynomial can have multiplicity 2, or one of them can have multiplicity 3 and the other multiplicity 1. The case when both have multiplicity 2 would imply that the polynomial is always greater than or equal to zero. So this case is impossible. Therefore, one of the two roots has multiplicity 3. Let' s denote this root by $\alpha$ . Then it is easy to see that $\alpha$ is a root of multiplicity 2 for $P'(x)=x^3+x^2+a$ and a root of multiplicity 1 for $P''(x)=3x^2+2x$ , so $\alpha=0$ or $\alpha=-\frac{2}{3}$ . Using that either $P'(0)=0$ or $P'(-\frac{2}{3})=0$ , we get that $a=0$ or $a=-\frac{4}{27}$ . Now using these two possible values of $a$ and the fact that $P(0)=0$ or $P(-\frac{2}{3})=0$ , respectively, we get that $b=0$ when $a=0$ or $b=-\frac{4}{81}$ when $a=-\frac{4}{27}$ . So the possible pairs $(a, b)$ are $(0, 0)$ or $(-\frac{4}{27}, -\frac{4}{81})$ . The maximum product $ab$ is attained at the second pair. Therefore $\frac{a}{b}= 3$ .