Find all $m\in\mathbb{R}$ such that $x^4+x^3-2x^2+3mx-m^2=0$ has only real roots.

Question

Find all $m\in\mathbb{R}$ such that $x^4+x^3-2x^2+3mx-m^2=0$ has only real roots.

I have to find $m$ algebraically but I didn't know how and tried to use calculus..

Let $f(x)=x^4+x^3-2x^2+3mx-m^2$. Using the first derivative $f'(x)=4x^3+3x^2-4x+3m$ and if we take $g(x)=4x^3+3x^2-4x=-3m$. But I think my idea is bad...

Answer 1

Hint. Note that the given polynomial can be factored as $$(x^2-x+m)(x^2+2x-m).$$ So it has only real roots if an only if $\Delta_1=1-4m\geq 0$ and $\Delta_2=4+4m\geq 0$.