Solve n:(z-1+i)^4-5(z-1+i)^2+4=0

$$n : ( z - 1 + i ) ^ { 4 } - 5 ( z - 1 + i ) ^ { 2 } + 4 = 0$$

Solve for n

$$n=\left(5\left(z+\left(-1+i\right)\right)^{2}-4\right)\left(z+\left(-1+i\right)\right)^{4}$$

$$z\neq 1-i$$

Steps for Solving Linear Equation

Multiply both sides of the equation by z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4.

$$\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(\frac{n}{\left(z-1+i\right)^{4}}-5\left(z-1+i\right)^{2}\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 4=0$$

Square z-1+i.

$$\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(\frac{n}{\left(z-1+i\right)^{4}}-5\left(z^{2}+\left(-2+2i\right)z-2i\right)\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 4=0$$

Use the distributive property to multiply -5 by z^{2}+\left(-2+2i\right)z-2i.

$$\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(\frac{n}{\left(z-1+i\right)^{4}}-5z^{2}+\left(10-10i\right)z+10i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 4=0$$

Use the distributive property to multiply z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 by \frac{n}{\left(z-1+i\right)^{4}}-5z^{2}+\left(10-10i\right)z+10i and combine like terms.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+150iz^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-200-200i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+300z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-120+120i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 4=0$$

Use the distributive property to multiply z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 by 4.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+150iz^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-200-200i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+300z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-120+120i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i+4z^{4}+\left(-16+16i\right)z^{3}-48iz^{2}+\left(32+32i\right)z-16=0$$

Combine 150iz^{4} and 4z^{4} to get \left(4+150i\right)z^{4}.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-200-200i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+300z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-120+120i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i+\left(-16+16i\right)z^{3}-48iz^{2}+\left(32+32i\right)z-16=0$$

Combine \left(-200-200i\right)z^{3} and \left(-16+16i\right)z^{3} to get \left(-216-184i\right)z^{3}.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+300z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-120+120i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-48iz^{2}+\left(32+32i\right)z-16=0$$

Combine 300z^{2} and -48iz^{2} to get \left(300-48i\right)z^{2}.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-120+120i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i+\left(32+32i\right)z-16=0$$

Combine \left(-120+120i\right)z and \left(32+32i\right)z to get \left(-88+152i\right)z.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}-5z^{6}+\left(30-30i\right)z^{5}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=0$$

Add 5z^{6} to both sides. Anything plus zero gives itself.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(30-30i\right)z^{5}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}$$

Subtract \left(30-30i\right)z^{5} from both sides.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(4+150i\right)z^{4}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}-\left(30-30i\right)z^{5}$$

Subtract \left(4+150i\right)z^{4} from both sides.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-216-184i\right)z^{3}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}$$

Subtract \left(-216-184i\right)z^{3} from both sides.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(300-48i\right)z^{2}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}$$

Subtract \left(300-48i\right)z^{2} from both sides.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-88+152i\right)z-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}$$

Subtract \left(-88+152i\right)z from both sides.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}-4\times \frac{n}{\left(z-1+i\right)^{4}}-40i-16=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z$$

Add 40i to both sides.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}-4\times \frac{n}{\left(z-1+i\right)^{4}}-16=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z+40i$$

Add 16 to both sides.

$$z^{4}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(-4+4i\right)z^{3}\times \frac{n}{\left(z-1+i\right)^{4}}-12iz^{2}\times \frac{n}{\left(z-1+i\right)^{4}}+\left(8+8i\right)z\times \frac{n}{\left(z-1+i\right)^{4}}-4\times \frac{n}{\left(z-1+i\right)^{4}}=5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z+40i+16$$

Multiply both sides of the equation by z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}-\left(30-30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Multiply -1 and 30-30i to get -30+30i.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}+\left(-30+30i\right)z^{5}-\left(4+150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Multiply -1 and 4+150i to get -4-150i.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}+\left(-30+30i\right)z^{5}+\left(-4-150i\right)z^{4}-\left(-216-184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Multiply -1 and -216-184i to get 216+184i.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}+\left(-30+30i\right)z^{5}+\left(-4-150i\right)z^{4}+\left(216+184i\right)z^{3}-\left(300-48i\right)z^{2}-\left(-88+152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Multiply -1 and 300-48i to get -300+48i.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}+\left(-30+30i\right)z^{5}+\left(-4-150i\right)z^{4}+\left(216+184i\right)z^{3}+\left(-300+48i\right)z^{2}-\left(-88+152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Multiply -1 and -88+152i to get 88-152i.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\left(5z^{6}+\left(-30+30i\right)z^{5}+\left(-4-150i\right)z^{4}+\left(216+184i\right)z^{3}+\left(-300+48i\right)z^{2}+\left(88-152i\right)z\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Use the distributive property to multiply z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 by 5z^{6}+\left(-30+30i\right)z^{5}+\left(-4-150i\right)z^{4}+\left(216+184i\right)z^{3}+\left(-300+48i\right)z^{2}+\left(88-152i\right)z and combine like terms.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8360i\right)z^{4}+\left(-5472-3808i\right)z^{3}+\left(3120-704i\right)z^{2}+\left(-352+608i\right)z+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times \left(40i\right)+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Use the distributive property to multiply z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 by 40i.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8360i\right)z^{4}+\left(-5472-3808i\right)z^{3}+\left(3120-704i\right)z^{2}+\left(-352+608i\right)z+40iz^{4}+\left(-160-160i\right)z^{3}+480z^{2}+\left(-320+320i\right)z-160i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Combine \left(1104+8360i\right)z^{4} and 40iz^{4} to get \left(1104+8400i\right)z^{4}.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8400i\right)z^{4}+\left(-5472-3808i\right)z^{3}+\left(3120-704i\right)z^{2}+\left(-352+608i\right)z+\left(-160-160i\right)z^{3}+480z^{2}+\left(-320+320i\right)z-160i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Combine \left(-5472-3808i\right)z^{3} and \left(-160-160i\right)z^{3} to get \left(-5632-3968i\right)z^{3}.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8400i\right)z^{4}+\left(-5632-3968i\right)z^{3}+\left(3120-704i\right)z^{2}+\left(-352+608i\right)z+480z^{2}+\left(-320+320i\right)z-160i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Combine \left(3120-704i\right)z^{2} and 480z^{2} to get \left(3600-704i\right)z^{2}.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8400i\right)z^{4}+\left(-5632-3968i\right)z^{3}+\left(3600-704i\right)z^{2}+\left(-352+608i\right)z+\left(-320+320i\right)z-160i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Combine \left(-352+608i\right)z and \left(-320+320i\right)z to get \left(-672+928i\right)z.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8400i\right)z^{4}+\left(-5632-3968i\right)z^{3}+\left(3600-704i\right)z^{2}+\left(-672+928i\right)z-160i+\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)\times 16$$

Use the distributive property to multiply z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 by 16.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1104+8400i\right)z^{4}+\left(-5632-3968i\right)z^{3}+\left(3600-704i\right)z^{2}+\left(-672+928i\right)z-160i+16z^{4}+\left(-64+64i\right)z^{3}-192iz^{2}+\left(128+128i\right)z-64$$

Combine \left(1104+8400i\right)z^{4} and 16z^{4} to get \left(1120+8400i\right)z^{4}.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5632-3968i\right)z^{3}+\left(3600-704i\right)z^{2}+\left(-672+928i\right)z-160i+\left(-64+64i\right)z^{3}-192iz^{2}+\left(128+128i\right)z-64$$

Combine \left(-5632-3968i\right)z^{3} and \left(-64+64i\right)z^{3} to get \left(-5696-3904i\right)z^{3}.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-704i\right)z^{2}+\left(-672+928i\right)z-160i-192iz^{2}+\left(128+128i\right)z-64$$

Combine \left(3600-704i\right)z^{2} and -192iz^{2} to get \left(3600-896i\right)z^{2}.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-672+928i\right)z-160i+\left(128+128i\right)z-64$$

Combine \left(-672+928i\right)z and \left(128+128i\right)z to get \left(-544+1056i\right)z.

$$z^{4}n+\left(-4+4i\right)z^{3}n-12iz^{2}n+\left(8+8i\right)zn-4n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-544+1056i\right)z-160i-64$$

Combine all terms containing n.

$$\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-544+1056i\right)z-160i-64$$

The equation is in standard form.

$$\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)n=5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-544+1056i\right)z+\left(-64-160i\right)$$

Divide both sides by z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4.

$$\frac{\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)n}{z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4}=\frac{5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-544+1056i\right)z+\left(-64-160i\right)}{z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4}$$

Dividing by z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4 undoes the multiplication by z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4.

$$n=\frac{5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-544+1056i\right)z+\left(-64-160i\right)}{z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4}$$

Divide 5z^{10}+\left(-50+50i\right)z^{9}+\left(-4-450i\right)z^{8}+\left(1232+1168i\right)z^{7}+\left(-544+1056i\right)z+\left(-4200+224i\right)z^{6}+\left(4592-5488i\right)z^{5}+\left(1120+8400i\right)z^{4}+\left(-5696-3904i\right)z^{3}+\left(3600-896i\right)z^{2}+\left(-64-160i\right) by z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4.

$$n=\left(5z^{2}+\left(-10+10i\right)z+\left(-4-10i\right)\right)\left(z^{4}+\left(-4+4i\right)z^{3}-12iz^{2}+\left(8+8i\right)z-4\right)$$