We have a polynomial $p(x)=x^5 -ax^3+b$. We need to find the relationship between $a$ and $b$ such that $p(x)$ has multiple roots.
Assume that $p(x)$ has 2 roots $c$ and $d$ with multiplicities $2$ and $3$ respectively. Then 1) $3c+2d=0$ 2) $3c^2+6cd+d^2=-a$ 3) $(c^3)*(d^2)= -b$
Thus, $a=3.75c^2$ and $b=-2.25c^5$. and we can find the relationship between $a$ and $b$ then.
Is it the right approach?