Solve f(x+5)*(3-x)=2x*(1-x)+12

$$f ( x + 5 ) \cdot ( 3 - x ) = 2 x \cdot ( 1 - x ) + 12$$

Solve for f (complex solution)

$$\left\{\begin{matrix}f=\frac{2\left(x+2\right)}{x+5}\text{, }&x\neq -5\\f\in \mathrm{C}\text{, }&x=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply f by x+5.

$$\left(fx+5f\right)\left(3-x\right)=2x\left(1-x\right)+12$$

Use the distributive property to multiply fx+5f by 3-x and combine like terms.

$$-2fx-fx^{2}+15f=2x\left(1-x\right)+12$$

Use the distributive property to multiply 2x by 1-x.

$$-2fx-fx^{2}+15f=2x-2x^{2}+12$$

Combine all terms containing f.

$$\left(-2x-x^{2}+15\right)f=2x-2x^{2}+12$$

The equation is in standard form.

$$\left(15-2x-x^{2}\right)f=12+2x-2x^{2}$$

Divide both sides by -2x-x^{2}+15.

$$\frac{\left(15-2x-x^{2}\right)f}{15-2x-x^{2}}=-\frac{2\left(x-3\right)\left(x+2\right)}{15-2x-x^{2}}$$

Dividing by -2x-x^{2}+15 undoes the multiplication by -2x-x^{2}+15.

$$f=-\frac{2\left(x-3\right)\left(x+2\right)}{15-2x-x^{2}}$$

Divide -2\left(-3+x\right)\left(2+x\right) by -2x-x^{2}+15.

$$f=\frac{2\left(x+2\right)}{x+5}$$

Solve for f

$$\left\{\begin{matrix}f=\frac{2\left(x+2\right)}{x+5}\text{, }&x\neq -5\\f\in \mathrm{R}\text{, }&x=3\end{matrix}\right.$$

Steps for Solving Linear Equation

Use the distributive property to multiply f by x+5.

$$\left(fx+5f\right)\left(3-x\right)=2x\left(1-x\right)+12$$

Use the distributive property to multiply fx+5f by 3-x and combine like terms.

$$-2fx-fx^{2}+15f=2x\left(1-x\right)+12$$

Use the distributive property to multiply 2x by 1-x.

$$-2fx-fx^{2}+15f=2x-2x^{2}+12$$

Combine all terms containing f.

$$\left(-2x-x^{2}+15\right)f=2x-2x^{2}+12$$

The equation is in standard form.

$$\left(15-2x-x^{2}\right)f=12+2x-2x^{2}$$

Divide both sides by -2x-x^{2}+15.

$$\frac{\left(15-2x-x^{2}\right)f}{15-2x-x^{2}}=-\frac{2\left(x-3\right)\left(x+2\right)}{15-2x-x^{2}}$$

Dividing by -2x-x^{2}+15 undoes the multiplication by -2x-x^{2}+15.

$$f=-\frac{2\left(x-3\right)\left(x+2\right)}{15-2x-x^{2}}$$

Divide -2\left(-3+x\right)\left(2+x\right) by -2x-x^{2}+15.

$$f=\frac{2\left(x+2\right)}{x+5}$$

Solve for x

$$\left\{\begin{matrix}\\x=3\text{, }&\text{unconditionally}\\x=-\frac{4-5f}{2-f}\text{, }&f\neq 2\end{matrix}\right.$$