Solve 1/(a+1)^3+x/a^2+a-1/a+1-x/a^2+2a+1=0

$$\frac { 1 } { ( a + 1 ) ^ { 3 } } + \frac { x } { a ^ { 2 } + a } - \frac { 1 } { a + 1 } - \frac { x } { a ^ { 2 } + 2 a + 1 } = 0$$

Solve for x

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

$$a\neq -1\text{ and }a\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by a\left(a+1\right)^{3}, the least common multiple of \left(a+1\right)^{3},a^{2}+a,a+1,a^{2}+2a+1.

$$a+\left(a+1\right)^{2}x-a\left(a+1\right)^{2}-a\left(a+1\right)x=0$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+1\right)^{2}.

$$a+\left(a^{2}+2a+1\right)x-a\left(a+1\right)^{2}-a\left(a+1\right)x=0$$

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

$$a+a^{2}x+2ax+x-a\left(a+1\right)^{2}-a\left(a+1\right)x=0$$

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(a+1\right)^{2}.

$$a+a^{2}x+2ax+x-a\left(a^{2}+2a+1\right)-a\left(a+1\right)x=0$$

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

$$a+a^{2}x+2ax+x-\left(a^{3}+2a^{2}+a\right)-a\left(a+1\right)x=0$$

To find the opposite of a^{3}+2a^{2}+a, find the opposite of each term.

$$a+a^{2}x+2ax+x-a^{3}-2a^{2}-a-a\left(a+1\right)x=0$$

Combine a and -a to get 0.

$$a^{2}x+2ax+x-a^{3}-2a^{2}-a\left(a+1\right)x=0$$

Use the distributive property to multiply a by a+1.

$$a^{2}x+2ax+x-a^{3}-2a^{2}-\left(a^{2}+a\right)x=0$$

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

$$a^{2}x+2ax+x-a^{3}-2a^{2}-\left(a^{2}x+ax\right)=0$$

To find the opposite of a^{2}x+ax, find the opposite of each term.

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

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

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

Combine 2ax and -ax to get ax.

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

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

$$ax+x-2a^{2}=a^{3}$$

Add 2a^{2} to both sides.

$$ax+x=a^{3}+2a^{2}$$

Combine all terms containing x.

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

Divide both sides by a+1.

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

Dividing by a+1 undoes the multiplication by a+1.

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