newton in maths!!! how

Using Newston's Sums method, we have:

$\begin{cases} r+s+t & = -3 \\ rs+st+tr & = 4 \\ rst & = 8 \end{cases}$

And:

$\begin{aligned} r^2 + s^2 + t^2 & = (r+s+t)^2 - 2(rs+st+tr) = (-3)^2 -2(4) = 1 \\ r^3 + s^3 + t^3 & = (r+s+t)(r^2 + s^2 + t^2) - (rs+st+tr)(r+s+t) \\ & \quad \quad + 3rst \\ & = (-3)(1)-4(-3)+3(8) = 33 \\ r^4 + s^4 + t^4 & = (r+s+t)(r^3 + s^3 + t^3) - (rs+st+tr)(r^2 + s^2 + t^2) \\ & \quad \quad + rst(r+s+t) \\ & = (-3)(33)-4(1)+8(-3) = \boxed{-127} \end{aligned}$