Converting -4 to polar form gives [tex]-4=4\exp(i\pi)[/tex].
Then the 4th roots of -4 would be the numbers
[tex]4^{1/4}\exp\left(i\dfrac{\pi+2k\pi}4\right)[/tex]
where k is taken from {0, 1, 2, 3}.
So we have
[tex]z_1=4^{1/4}\exp\left(\dfrac{i\pi}4\right)=\sqrt2\left(\cos\dfrac\pi4+i\sin\dfrac\pi4\right)=1+i[/tex]
[tex]z_2=\sqrt2\left(\cos\dfrac{3\pi}4+i\sin\dfrac{3\pi}4\right)=-1+i[/tex]
[tex]z_3=\sqrt2\left(\cos\dfrac{5\pi}4+i\sin\dfrac{5\pi}4\right)=-1-i[/tex]
[tex]z_4=\sqrt2\left(\cos\dfrac{7\pi}4+i\sin\dfrac{7\pi}4\right)=1-i[/tex]
