Solve 4a^3x-4a^2x=(a-1)(x+2a+1)

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

Solve for x (complex solution)

$$\left\{\begin{matrix}x=\frac{1}{2a-1}\text{, }&a\neq \frac{1}{2}\\x\in \mathrm{C}\text{, }&a=-\frac{1}{2}\text{ or }a=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-1 by x+2a+1 and combine like terms.

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

Subtract ax from both sides.

$$4a^{3}x-4a^{2}x-ax=2a^{2}-a-x-1$$

Add x to both sides.

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

Combine all terms containing x.

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

Divide both sides by 4a^{3}-4a^{2}-a+1.

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

Dividing by 4a^{3}-4a^{2}-a+1 undoes the multiplication by 4a^{3}-4a^{2}-a+1.

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

Divide \left(-1+a\right)\left(1+2a\right) by 4a^{3}-4a^{2}-a+1.

$$x=\frac{1}{2a-1}$$

Solve for x

$$\left\{\begin{matrix}x=\frac{1}{2a-1}\text{, }&a\neq \frac{1}{2}\\x\in \mathrm{R}\text{, }&a=-\frac{1}{2}\text{ or }a=1\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply a-1 by x+2a+1 and combine like terms.

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

Subtract ax from both sides.

$$4a^{3}x-4a^{2}x-ax=2a^{2}-a-x-1$$

Add x to both sides.

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

Combine all terms containing x.

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

Divide both sides by 4a^{3}-4a^{2}-a+1.

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

Dividing by 4a^{3}-4a^{2}-a+1 undoes the multiplication by 4a^{3}-4a^{2}-a+1.

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

Divide \left(-1+a\right)\left(1+2a\right) by 4a^{3}-4a^{2}-a+1.

$$x=\frac{1}{2a-1}$$

Solve for a

$$\left\{\begin{matrix}\\a=-\frac{1}{2}=-0.5\text{; }a=1\text{, }&\text{unconditionally}\\a=\frac{1}{2}+\frac{1}{2x}\text{, }&x\neq 0\end{matrix}\right.$$