Simplify (x+2)/(x^2-2x-15)+x/(x+3)

$\frac{x + 2}{x^{2} - 2 x - 15} + \frac{x}{x + 3}$

Step 1

Factor $x^{2} - 2 x - 15$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 15$ and whose sum is $- 2$ .

$- 5, 3$

Write the factored form using these integers.

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

Step 2

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

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

Step 3

Write each expression with a common denominator of $(x - 5) (x + 3)$ , by multiplying each by an appropriate factor of $1$ .

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

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

Reorder the factors of $(x - 5) (x + 3)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Move $- 5$ to the left of $x$ .

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

Subtract $5 x$ from $x$ .

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

Reorder terms.

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

Simplify (x+2)/(x^2-2x-15)+x/(x+3)

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