Solve x^5+x^4=(x^2+ax+1)(x^3-6x+1)

$$x ^ { 5 } + x ^ { 4 } = ( x ^ { 2 } + a x + 1 ) ( x ^ { 3 } - 6 x + 1 )$$

Solve for a

$$a=-\frac{1-6x+x^{2}-5x^{3}-x^{4}}{x\left(x^{3}-6x+1\right)}$$

$$x\neq \sqrt{2}\left(-\sqrt{3}\sin(\frac{\arccos(\frac{\sqrt{2}}{8})}{3})+\cos(\frac{\arccos(\frac{\sqrt{2}}{8})}{3})\right)\text{ and }x\neq -2\sqrt{2}\cos(\frac{\arccos(\frac{\sqrt{2}}{8})}{3})\text{ and }x\neq \sqrt{2}\left(\sqrt{3}\sin(\frac{\arccos(\frac{\sqrt{2}}{8})}{3})+\cos(\frac{\arccos(\frac{\sqrt{2}}{8})}{3})\right)\text{ and }x\neq 0$$

Steps for Solving Linear Equation

Use the distributive property to multiply x^{2}+ax+1 by x^{3}-6x+1 and combine like terms.

$$x^{5}+x^{4}=x^{5}-5x^{3}+x^{2}+ax^{4}-6ax^{2}+ax-6x+1$$

Swap sides so that all variable terms are on the left hand side.

$$x^{5}-5x^{3}+x^{2}+ax^{4}-6ax^{2}+ax-6x+1=x^{5}+x^{4}$$

Subtract x^{5} from both sides.

$$-5x^{3}+x^{2}+ax^{4}-6ax^{2}+ax-6x+1=x^{5}+x^{4}-x^{5}$$

Combine x^{5} and -x^{5} to get 0.

$$-5x^{3}+x^{2}+ax^{4}-6ax^{2}+ax-6x+1=x^{4}$$

Add 5x^{3} to both sides.

$$x^{2}+ax^{4}-6ax^{2}+ax-6x+1=x^{4}+5x^{3}$$

Subtract x^{2} from both sides.

$$ax^{4}-6ax^{2}+ax-6x+1=x^{4}+5x^{3}-x^{2}$$

Add 6x to both sides.

$$ax^{4}-6ax^{2}+ax+1=x^{4}+5x^{3}-x^{2}+6x$$

Subtract 1 from both sides.

$$ax^{4}-6ax^{2}+ax=x^{4}+5x^{3}-x^{2}+6x-1$$

Combine all terms containing a.

$$\left(x^{4}-6x^{2}+x\right)a=x^{4}+5x^{3}-x^{2}+6x-1$$

Divide both sides by x^{4}-6x^{2}+x.

$$\frac{\left(x^{4}-6x^{2}+x\right)a}{x^{4}-6x^{2}+x}=\frac{x^{4}+5x^{3}-x^{2}+6x-1}{x^{4}-6x^{2}+x}$$

Dividing by x^{4}-6x^{2}+x undoes the multiplication by x^{4}-6x^{2}+x.

$$a=\frac{x^{4}+5x^{3}-x^{2}+6x-1}{x^{4}-6x^{2}+x}$$

Divide x^{4}+5x^{3}-x^{2}+6x-1 by x^{4}-6x^{2}+x.

$$a=\frac{x^{4}+5x^{3}-x^{2}+6x-1}{x\left(x^{3}-6x+1\right)}$$