### Instruction: Find the solutions to $z^4 = -4.$  Enter the solutions, separated by commas.### Response: This is a complex number equation, so I need to use the polar form of complex numbers to solve it.
First, I'll rewrite -4 as $4e^{i\pi}$, using Euler's formula and the fact that -1 has an argument of $\pi$.
Then, I'll use the fourth root property to find the four solutions to $z^4 = 4e^{i\pi}$.
This means that $z = \sqrt[4]{4}e^{i(\pi/4 + k\pi/2)}$, where $k = 0, 1, 2, 3$.
Simplifying, I get $z = \sqrt{2}e^{i(\pi/4 + k\pi/2)}$.
To find the Cartesian form of each solution, I'll use Euler's formula again and expand the exponentials using the sum of angles formula.
For $k = 0$, I get $z = \sqrt{2}e^{i\pi/4} = \sqrt{2}(\cos(\pi/4) + i\sin(\pi/4)) = 1 + i$.
For $k = 1$, I get $z = \sqrt{2}e^{i3\pi/4} = \sqrt{2}(\cos(3\pi/4) + i\sin(3\pi/4)) = -1 + i$.
For $k = 2$, I get $z = \sqrt{2}e^{i5\pi/4} = \sqrt{2}(\cos(5\pi/4) + i\sin(5\pi/4)) = -1 - i$.
For $k = 3$, I get $z = \sqrt{2}e^{i7\pi/4} = \sqrt{2}(\cos(7\pi/4) + i\sin(7\pi/4)) = 1 - i$.