Solve 2x/a^3-8-a/a^2+2a+4=x-1/a-2

$$\frac { 2 x } { a ^ { 3 } - 8 } - \frac { a } { a ^ { 2 } + 2 a + 4 } = \frac { x - 1 } { a - 2 }$$

Solve for x

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

$$a\neq 2$$

Steps for Solving Linear Equation

Multiply both sides of the equation by \left(a-2\right)\left(a^{2}+2a+4\right), the least common multiple of a^{3}-8,a^{2}+2a+4,a-2.

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

Use the distributive property to multiply a-2 by a.

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

To find the opposite of a^{2}-2a, find the opposite of each term.

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

Use the distributive property to multiply a^{2}+2a+4 by x-1.

$$2x-a^{2}+2a=a^{2}x-a^{2}+2ax-2a+4x-4$$

Subtract a^{2}x from both sides.

$$2x-a^{2}+2a-a^{2}x=-a^{2}+2ax-2a+4x-4$$

Subtract 2ax from both sides.

$$2x-a^{2}+2a-a^{2}x-2ax=-a^{2}-2a+4x-4$$

Subtract 4x from both sides.

$$2x-a^{2}+2a-a^{2}x-2ax-4x=-a^{2}-2a-4$$

Combine 2x and -4x to get -2x.

$$-2x-a^{2}+2a-a^{2}x-2ax=-a^{2}-2a-4$$

Add a^{2} to both sides.

$$-2x+2a-a^{2}x-2ax=-a^{2}-2a-4+a^{2}$$

Combine -a^{2} and a^{2} to get 0.

$$-2x+2a-a^{2}x-2ax=-2a-4$$

Subtract 2a from both sides.

$$-2x-a^{2}x-2ax=-2a-4-2a$$

Combine -2a and -2a to get -4a.

$$-2x-a^{2}x-2ax=-4a-4$$

Combine all terms containing x.

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

The equation is in standard form.

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

Divide both sides by -2-a^{2}-2a.

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

Dividing by -2-a^{2}-2a undoes the multiplication by -2-a^{2}-2a.

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

Divide -4a-4 by -2-a^{2}-2a.

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

Solve for a

$$\left\{\begin{matrix}a=\frac{\sqrt{4-x^{2}}-x+2}{x}\text{, }&x\neq \frac{6}{5}\text{ and }x\neq 0\text{ and }|x|\leq 2\\a=-\frac{\sqrt{4-x^{2}}+x-2}{x}\text{, }&x\neq 0\text{ and }|x|\leq 2\\a=-1\text{, }&x=0\end{matrix}\right.$$