Find All Complex Solutions (10x+25)/(3x+12)=(5x)/(x+4)

Factor ${\displaystyle 5}$ out of ${\displaystyle 10x+25}$.

Factor ${\displaystyle 3}$ out of ${\displaystyle 3x+12}$.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

The LCM is the smallest positive number that all of the numbers divide into evenly.

Since ${\displaystyle 3}$ has no factors besides ${\displaystyle 1}$ and ${\displaystyle 3}$.

The number ${\displaystyle 1}$ is not a prime number because it only has one positive factor, which is itself.

The LCM of ${\displaystyle 3,1}$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

The factor for ${\displaystyle x+4}$ is ${\displaystyle x+4}$ itself.

The LCM of ${\displaystyle x+4,x+4}$ is the result of multiplying all factors the greatest number of times they occur in either term.

The Least Common Multiple ${\displaystyle \mathrm{LCM}}$ of some numbers is the smallest number that the numbers are factors of.

Multiply each term in ${\displaystyle \frac{5(2x+5)}{3(x+4)}=\frac{5x}{x+4}}$ by ${\displaystyle 3(x+4)}$.

Simplify the left side.

Simplify the right side.

Move all terms containing ${\displaystyle x}$ to the left side of the equation.

Subtract ${\displaystyle 25}$ from both sides of the equation.

Divide each term in ${\displaystyle -5x=-25}$ by ${\displaystyle -5}$ and simplify.