Solve (a-2)x^3+ax^n+4x+2=0

$$( a - 2 ) x ^ { 3 } + a x ^ { n } + 4 x + 2 = 0$$

Solve for a

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

$$x>0\text{ or }\left(Denominator(n)\text{bmod}2=1\text{ and }x<0\right)$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-2 by x^{3}.

$$ax^{3}-2x^{3}+ax^{n}+4x+2=0$$

Add 2x^{3} to both sides. Anything plus zero gives itself.

$$ax^{3}+ax^{n}+4x+2=2x^{3}$$

Subtract 4x from both sides.

$$ax^{3}+ax^{n}+2=2x^{3}-4x$$

Subtract 2 from both sides.

$$ax^{3}+ax^{n}=2x^{3}-4x-2$$

Combine all terms containing a.

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

Divide both sides by x^{3}+x^{n}.

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

Dividing by x^{3}+x^{n} undoes the multiplication by x^{3}+x^{n}.

$$a=\frac{2x^{3}-4x-2}{x^{3}+x^{n}}$$

Divide 2x^{3}-4x-2 by x^{3}+x^{n}.

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

Solve for n

$$\left\{\begin{matrix}n=\log_{x}\left(-\frac{ax^{3}-2x^{3}+4x+2}{a}\right)\text{, }&\left(a>-\frac{2+4x-2x^{3}}{x^{3}}\text{ and }a<0\text{ and }x\neq 1\text{ and }x>0\right)\text{ or }\left(a<-\frac{2+4x-2x^{3}}{x^{3}}\text{ and }a>0\text{ and }x\neq 1\text{ and }x>0\right)\\n\in \mathrm{R}\text{, }&\left(x=\frac{\sqrt{5}+1}{2}\text{ and }a=0\right)\text{ or }\left(x=1\text{ and }a=-2\right)\text{ or }\left(x=-1\text{ and }a=0\text{ and }Denominator(n)\text{bmod}2=1\right)\text{ or }\left(x=\frac{1-\sqrt{5}}{2}\text{ and }a=0\text{ and }Denominator(n)\text{bmod}2=1\right)\end{matrix}\right.$$