$z_1+z_2=5$

$z_1^3+z_2^3=20+15 i$

$z_1^3+z_2^3=\left(z_1+z_2\right)^3-3 z_1 z_2\left(z_1+z_2\right)$

$z_1^3+z_2^3=125-3 z_1 \cdot z_2(5)$

$\Rightarrow 20+15 i=125-15 z_1 z_2$

$\Rightarrow 3 z_1 z_2=25-4-3 i$

$\Rightarrow 3 z_1 z_2=21-3 i$

$\Rightarrow z_1 \cdot z_2=7-i$

$\Rightarrow\left(z_1+z_2\right)^2=25$

$\Rightarrow z_1^2+z_2^2=25-2(7-i)$

$\Rightarrow 11+2 i$

$\left(z_1^2+z_2^2\right)^2=121-4+44 i$

$\Rightarrow z_1^4+z_2^4+2(7-i)^2=117+44 i$

$\Rightarrow z_1^4+z_2^4=117+44 i-2(49-1-14 i)$

$\Rightarrow\left|z_1^4+z_2^4\right|=75$