I'm stuck on this algebra question

The solutions are 1 + sqrt(3)*i, 1 - sqrt(3)*i, -1 + sqrt(3)*i, -1 - sqrt(3)*i

It is a power of 4 so there are 4 solutions

this is one way to tackle it
\(x^4=-64\\ x^2=8i\qquad or \qquad x^2=-8i\\ x=\pm\sqrt 8 [\sqrt i] \qquad or \qquad x \pm\sqrt 8\sqrt{-1} [\sqrt i] \\ x=\pm\sqrt 8 [\sqrt i] \qquad or \qquad x \pm i\sqrt 8 [\sqrt i] \\ x=\pm2\sqrt 2 [\sqrt i] \qquad or \qquad x \pm2 i\sqrt 2 [\sqrt i] \\ \)

Now you need to check what I have done and work out what the square root of  i  is.

there is lots of places on the internet help you to learn how to work this out.

Here is one:

https://www.math.toronto.edu/mathnet/questionCorner/rootofi.html