If $\displaystyle \alpha$ and $\displaystyle \beta$ are roots of the equation $\displaystyle x^2+mx+n=0$, find the roots of $\displaystyle nx^2+(2n-m^2)x+n=0$ in terms of $\displaystyle \alpha$ and $\displaystyle \beta$
From the first equation you can get n and m in terms of $\displaystyle \alpha$ and $\displaystyle \beta$:
$\displaystyle \alpha \, \beta = n$.
$\displaystyle \alpha + \beta = -m$.
Solve the second equation using the quadratic formula. Then replace n and m in terms of $\displaystyle \alpha$ and $\displaystyle \beta$.
Sorry, but messy in maths is like digging a hole with a spade. It can be done but it takes effort. Your effort, not mine.
If you're genuinely stuck (as in you have absolutley no idea what to do) I or someone else will give additional assistance (but not with the manual labour).
But, instead of a spade you might try using a bobcat by noting that if $\displaystyle \gamma$ and $\displaystyle \delta$ are roots of the second equation, then
$\displaystyle \gamma \, \delta = 1$
$\displaystyle \gamma + \delta = \frac{m^2}{n} - 2$
where I've obviously divided the second equation through by n first .....