(vii) z4+5z2+6z^4 + 5z^2 + 6 z4+5z2+6 Solution Let u=z2:u2+5u+6=(u+2)(u+3)So:z4+5z2+6=(z2+2)(z2+3)Factor each:z2+2=(z−i2)(z+i2)z2+3=(z−i3)(z+i3)Therefore:z4+5z2+6=(z−i2)(z+i2)(z−i3)(z+i3)\begin{aligned} & \boxed{\text{Let } u = z^2:} \\ \\ & u^2 + 5u + 6 = (u + 2)(u + 3) \\ \\ & \boxed{\text{So:}} \\ & z^4 + 5z^2 + 6 = (z^2 + 2)(z^2 + 3) \\ \\ & \boxed{\text{Factor each:}} \\ & z^2 + 2 = (z - i\sqrt{2})(z + i\sqrt{2}) \\ & z^2 + 3 = (z - i\sqrt{3})(z + i\sqrt{3}) \\ \\ & \boxed{\text{Therefore:}} \\ & z^4 + 5z^2 + 6 = (z - i\sqrt{2})(z + i\sqrt{2})(z - i\sqrt{3})(z + i\sqrt{3}) \end{aligned}Let u=z2:u2+5u+6=(u+2)(u+3)So:z4+5z2+6=(z2+2)(z2+3)Factor each:z2+2=(z−i2)(z+i2)z2+3=(z−i3)(z+i3)Therefore:z4+5z2+6=(z−i2)(z+i2)(z−i3)(z+i3)