If $\displaystyle (x - a)^2$ is a factor of $\displaystyle x^3 + mx^2 + n$, prove that $\displaystyle 27n + 4m^3 = 0$
Let $\displaystyle f(x)=x^3+mx^2+n$
If $\displaystyle (x-a)^2$ is a factor of f, then $\displaystyle f(a)=0, \ f'(a)=0$
$\displaystyle f(a)=0\Rightarrow a^3+ma^2+n=0$
$\displaystyle f'(a)=0\Rightarrow 3a^2+2am=0\Rightarrow a(3a+2m)=0$
If $\displaystyle a\neq 0\Rightarrow a=-\frac{2m}{3}$
Now replace a in $\displaystyle f(a)=0$