jmc

The distance between $(1+2i)z^3$ and $z^5$ is $$\begin{array}{rl}|(1+2i)z^3-z^5|& =|z^3|\cdot |(1+2i)-z^2|\\ & =5^3\cdot |(1+2i)-z^2|,\end{array}$$since we are given $|z|=5.$ We have $|z^2|=25;$ that is, in the complex plane, $z^2$ lies on the circle centered at $0$ of radius $25.$ Given this fact, to maximize the distance from $z^2$ to $1+2i,$ we should choose $z^2$ to be a negative multiple of $1+2i$ (on the "opposite side" of $1+2i$ relative to the origin $0$). Since $|1+2i|=\sqrt{5}$ and $z^2$ must have magnitude $25$, scaling $1+2i$ by a factor of $-\frac{25}{\sqrt{5}}=-5\sqrt{5}$ gives the correct point: $$z^2=-5\sqrt{5}(1+2i).$$Then $$z^4=125(-3+4i)=\boxed{-375+500i}.$$(Note that the restriction $b>0$ was not used. It is only needed to ensure that the number $z$ in the problem statement is uniquely determined, since there are two complex numbers $z$ with $|z|=5$ such that $|(1+2i)z^3-z^5|$ is maximized, one the negation of the other.)