Let $$z_1$$ and $$z_2$$ be two complex numbers such that $$z_1 + z_2 = 5$$ and $$z_1^3 + z_2^3 = 20 + 15i$$. Then $$|z_1^4 + z_2^4|$$ equals

Use $$z_1^3 + z_2^3 = (z_1 + z_2)( (z_1+z_2)^2 - 3z_1z_2 )$$
$$20 + 15i = 5( 25 - 3z_1z_2 ) \implies 4 + 3i = 25 - 3z_1z_2$$
$$3z_1z_2 = 21 - 3i \implies z_1z_2 = 7 - i$$.
2. Now find $$z_1^2 + z_2^2 = (z_1 + z_2)^2 - 2z_1z_2 = 25 - 2(7 - i) = 25 - 14 + 2i = 11 + 2i$$.
3. Find $$z_1^4 + z_2^4 = (z_1^2 + z_2^2)^2 - 2(z_1z_2)^2$$
$$z_1^4 + z_2^4 = (11 + 2i)^2 - 2(7 - i)^2$$
$$= (121 - 4 + 44i) - 2(49 - 1 - 14i)$$
$$= (117 + 44i) - 2(48 - 14i)$$
$$= 117 + 44i - 96 + 28i = 21 + 72i$$

$$z_1^4+z_2^4=21+72i$$

|21+72i|
=$$\sqrt{21^2+72^2}$$
=$$\sqrt{5625}=75$$