Simplify (5x)/(x^2-9)+2/(x+4)

$\frac{5 x}{x^{2} - 9} + \frac{2}{x + 4}$

Step 1

Simplify the denominator.

Rewrite $9$ as $3^{2}$ .

$\frac{5 x}{x^{2} - 3^{2}} + \frac{2}{x + 4}$

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 3$ .

$\frac{5 x}{(x + 3) (x - 3)} + \frac{2}{x + 4}$

Step 2

To write $\frac{5 x}{(x + 3) (x - 3)}$ as a fraction with a common denominator, multiply by $\frac{x + 4}{x + 4}$ .

$\frac{5 x}{(x + 3) (x - 3)} \cdot \frac{x + 4}{x + 4} + \frac{2}{x + 4}$

Step 3

To write $\frac{2}{x + 4}$ as a fraction with a common denominator, multiply by $\frac{(x + 3) (x - 3)}{(x + 3) (x - 3)}$ .

$\frac{5 x}{(x + 3) (x - 3)} \cdot \frac{x + 4}{x + 4} + \frac{2}{x + 4} \cdot \frac{(x + 3) (x - 3)}{(x + 3) (x - 3)}$

Step 4

Write each expression with a common denominator of $(x + 3) (x - 3) (x + 4)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5 x}{(x + 3) (x - 3)}$ by $\frac{x + 4}{x + 4}$ .

$\frac{5 x (x + 4)}{(x + 3) (x - 3) (x + 4)} + \frac{2}{x + 4} \cdot \frac{(x + 3) (x - 3)}{(x + 3) (x - 3)}$

Multiply $\frac{2}{x + 4}$ by $\frac{(x + 3) (x - 3)}{(x + 3) (x - 3)}$ .

$\frac{5 x (x + 4)}{(x + 3) (x - 3) (x + 4)} + \frac{2 ((x + 3) (x - 3))}{(x + 4) ((x + 3) (x - 3))}$

Reorder the factors of $(x + 3) (x - 3) (x + 4)$ .

$\frac{5 x (x + 4)}{(x + 3) (x + 4) (x - 3)} + \frac{2 ((x + 3) (x - 3))}{(x + 4) ((x + 3) (x - 3))}$

Reorder the factors of $(x + 4) ((x + 3) (x - 3))$ .

$\frac{5 x (x + 4)}{(x + 3) (x + 4) (x - 3)} + \frac{2 ((x + 3) (x - 3))}{(x + 3) (x + 4) (x - 3)}$

Step 5

Combine the numerators over the common denominator.

$\frac{5 x (x + 4) + 2 ((x + 3) (x - 3))}{(x + 3) (x + 4) (x - 3)}$

Step 6

Simplify the numerator.

Apply the distributive property.

$\frac{5 x \cdot x + 5 x \cdot 4 + 2 (x + 3) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{5 (x \cdot x) + 5 x \cdot 4 + 2 (x + 3) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Multiply $x$ by $x$ .

$\frac{5 x^{2} + 5 x \cdot 4 + 2 (x + 3) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Multiply $4$ by $5$ .

$\frac{5 x^{2} + 20 x + 2 (x + 3) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Apply the distributive property.

$\frac{5 x^{2} + 20 x + (2 x + 2 \cdot 3) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Multiply $2$ by $3$ .

$\frac{5 x^{2} + 20 x + (2 x + 6) (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Expand $(2 x + 6) (x - 3)$ using the FOIL Method.

Apply the distributive property.

$\frac{5 x^{2} + 20 x + 2 x (x - 3) + 6 (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Apply the distributive property.

$\frac{5 x^{2} + 20 x + 2 x \cdot x + 2 x \cdot - 3 + 6 (x - 3)}{(x + 3) (x + 4) (x - 3)}$

Apply the distributive property.

$\frac{5 x^{2} + 20 x + 2 x \cdot x + 2 x \cdot - 3 + 6 x + 6 \cdot - 3}{(x + 3) (x + 4) (x - 3)}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$\frac{5 x^{2} + 20 x + 2 (x \cdot x) + 2 x \cdot - 3 + 6 x + 6 \cdot - 3}{(x + 3) (x + 4) (x - 3)}$

Multiply $x$ by $x$ .

$\frac{5 x^{2} + 20 x + 2 x^{2} + 2 x \cdot - 3 + 6 x + 6 \cdot - 3}{(x + 3) (x + 4) (x - 3)}$

Multiply $- 3$ by $2$ .

$\frac{5 x^{2} + 20 x + 2 x^{2} - 6 x + 6 x + 6 \cdot - 3}{(x + 3) (x + 4) (x - 3)}$

Multiply $6$ by $- 3$ .

$\frac{5 x^{2} + 20 x + 2 x^{2} - 6 x + 6 x - 18}{(x + 3) (x + 4) (x - 3)}$

Add $- 6 x$ and $6 x$ .

$\frac{5 x^{2} + 20 x + 2 x^{2} + 0 - 18}{(x + 3) (x + 4) (x - 3)}$

Add $2 x^{2}$ and $0$ .

$\frac{5 x^{2} + 20 x + 2 x^{2} - 18}{(x + 3) (x + 4) (x - 3)}$

Add $5 x^{2}$ and $2 x^{2}$ .

$\frac{7 x^{2} + 20 x - 18}{(x + 3) (x + 4) (x - 3)}$

Simplify (5x)/(x^2-9)+2/(x+4)

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