Simplify (3x)/(x+2)+10/(x-2)-24/(x^2-4)

Rewrite ${\displaystyle 4}$ as ${\displaystyle 2^2}$.

Since both terms are perfect squares, factor using the difference of squares formula, ${\displaystyle a^2-b^2=(a+b)(a-b)}$ where ${\displaystyle a=x}$ and ${\displaystyle b=2}$.

Multiply ${\displaystyle \frac{3x}{x+2}}$ by ${\displaystyle \frac{x-2}{x-2}}$.

Multiply ${\displaystyle \frac{10}{x-2}}$ by ${\displaystyle \frac{x+2}{x+2}}$.

Reorder the factors of ${\displaystyle (x-2)(x+2)}$.

Combine the numerators over the common denominator.

Combine the numerators over the common denominator.

Apply the distributive property.

Multiply ${\displaystyle x}$ by ${\displaystyle x}$ by adding the exponents.

Multiply ${\displaystyle -2}$ by ${\displaystyle 3}$.

Apply the distributive property.

Multiply ${\displaystyle 10}$ by ${\displaystyle 2}$.

Add ${\displaystyle -6x}$ and ${\displaystyle 10x}$.

Subtract ${\displaystyle 24}$ from ${\displaystyle 20}$.

For a polynomial of the form ${\displaystyle a{x}^2+bx+c}$, rewrite the middle term as a sum of two terms whose product is ${\displaystyle a\cdot c=3\cdot -4=-12}$ and whose sum is ${\displaystyle b=4}$.

Factor out the greatest common factor from each group.

Factor the polynomial by factoring out the greatest common factor, ${\displaystyle 3x-2}$.

Cancel the common factor.

Rewrite the expression.