Solve f(3x)-f(x-1)=4(x-1)(2x+1)

$$f ( 3 x ) - f ( x - 1 ) = 4 ( x - 1 ) ( 2 x + 1 )$$

Solve for f (complex solution)

$$\left\{\begin{matrix}\\f=4\left(x-1\right)\text{, }&\text{unconditionally}\\f\in \mathrm{C}\text{, }&x=-\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply f by x-1.

$$f\times 3x-\left(fx-f\right)=4\left(x-1\right)\left(2x+1\right)$$

To find the opposite of fx-f, find the opposite of each term.

$$f\times 3x-fx+f=4\left(x-1\right)\left(2x+1\right)$$

Combine f\times 3x and -fx to get 2fx.

$$2fx+f=4\left(x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply 4 by x-1.

$$2fx+f=\left(4x-4\right)\left(2x+1\right)$$

Use the distributive property to multiply 4x-4 by 2x+1 and combine like terms.

$$2fx+f=8x^{2}-4x-4$$

Combine all terms containing f.

$$\left(2x+1\right)f=8x^{2}-4x-4$$

Divide both sides by 1+2x.

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

Dividing by 1+2x undoes the multiplication by 1+2x.

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

Divide 4\left(-1+x\right)\left(1+2x\right) by 1+2x.

$$f=4x-4$$

Solve for f

$$\left\{\begin{matrix}\\f=4\left(x-1\right)\text{, }&\text{unconditionally}\\f\in \mathrm{R}\text{, }&x=-\frac{1}{2}\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply f by x-1.

$$f\times 3x-\left(fx-f\right)=4\left(x-1\right)\left(2x+1\right)$$

To find the opposite of fx-f, find the opposite of each term.

$$f\times 3x-fx+f=4\left(x-1\right)\left(2x+1\right)$$

Combine f\times 3x and -fx to get 2fx.

$$2fx+f=4\left(x-1\right)\left(2x+1\right)$$

Use the distributive property to multiply 4 by x-1.

$$2fx+f=\left(4x-4\right)\left(2x+1\right)$$

Use the distributive property to multiply 4x-4 by 2x+1 and combine like terms.

$$2fx+f=8x^{2}-4x-4$$

Combine all terms containing f.

$$\left(2x+1\right)f=8x^{2}-4x-4$$

Divide both sides by 1+2x.

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

Dividing by 1+2x undoes the multiplication by 1+2x.

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

Divide 4\left(-1+x\right)\left(1+2x\right) by 1+2x.

$$f=4x-4$$

Solve for x

$$x=\frac{f+4}{4}$$

$$x = -\frac{1}{2} = -0.5$$