Simplify (x+6)/(x^2+8x+15)+(3x)/(x+5)-(x-3)/(x+3)

$\frac{x + 6}{x^{2} + 8 x + 15} + \frac{3 x}{x + 5} - \frac{x - 3}{x + 3}$

Step 1

Factor $x^{2} + 8 x + 15$ 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 $8$ .

$3, 5$

Write the factored form using these integers.

$\frac{x + 6}{(x + 3) (x + 5)} + \frac{3 x}{x + 5} - \frac{x - 3}{x + 3}$

Step 2

Find the common denominator.

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

$\frac{x + 6}{(x + 3) (x + 5)} + \frac{3 x}{x + 5} \cdot \frac{x + 3}{x + 3} - \frac{x - 3}{x + 3}$

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

$\frac{x + 6}{(x + 3) (x + 5)} + \frac{3 x (x + 3)}{(x + 5) (x + 3)} - \frac{x - 3}{x + 3}$

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

$\frac{x + 6}{(x + 3) (x + 5)} + \frac{3 x (x + 3)}{(x + 5) (x + 3)} - (\frac{x - 3}{x + 3} \cdot \frac{x + 5}{x + 5})$

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

$\frac{x + 6}{(x + 3) (x + 5)} + \frac{3 x (x + 3)}{(x + 5) (x + 3)} - \frac{(x - 3) (x + 5)}{(x + 3) (x + 5)}$

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

$\frac{x + 6}{(x + 3) (x + 5)} + \frac{3 x (x + 3)}{(x + 3) (x + 5)} - \frac{(x - 3) (x + 5)}{(x + 3) (x + 5)}$

Step 3

Combine the numerators over the common denominator.

$\frac{x + 6 + 3 x (x + 3) - (x - 3) (x + 5)}{(x + 3) (x + 5)}$

Step 4

Simplify each term.

Apply the distributive property.

$\frac{x + 6 + 3 x \cdot x + 3 x \cdot 3 - (x - 3) (x + 5)}{(x + 3) (x + 5)}$

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

Move $x$ .

$\frac{x + 6 + 3 (x \cdot x) + 3 x \cdot 3 - (x - 3) (x + 5)}{(x + 3) (x + 5)}$

Multiply $x$ by $x$ .

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

Multiply $3$ by $3$ .

$\frac{x + 6 + 3 x^{2} + 9 x - (x - 3) (x + 5)}{(x + 3) (x + 5)}$

Apply the distributive property.

$\frac{x + 6 + 3 x^{2} + 9 x + (- x - - 3) (x + 5)}{(x + 3) (x + 5)}$

Multiply $- 1$ by $- 3$ .

$\frac{x + 6 + 3 x^{2} + 9 x + (- x + 3) (x + 5)}{(x + 3) (x + 5)}$

Expand $(- x + 3) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$\frac{x + 6 + 3 x^{2} + 9 x - x (x + 5) + 3 (x + 5)}{(x + 3) (x + 5)}$

Apply the distributive property.

$\frac{x + 6 + 3 x^{2} + 9 x - x \cdot x - x \cdot 5 + 3 (x + 5)}{(x + 3) (x + 5)}$

Apply the distributive property.

$\frac{x + 6 + 3 x^{2} + 9 x - x \cdot x - x \cdot 5 + 3 x + 3 \cdot 5}{(x + 3) (x + 5)}$

Simplify and combine like terms.

Simplify each term.

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

Move $x$ .

$\frac{x + 6 + 3 x^{2} + 9 x - (x \cdot x) - x \cdot 5 + 3 x + 3 \cdot 5}{(x + 3) (x + 5)}$

Multiply $x$ by $x$ .

$\frac{x + 6 + 3 x^{2} + 9 x - x^{2} - x \cdot 5 + 3 x + 3 \cdot 5}{(x + 3) (x + 5)}$

Multiply $5$ by $- 1$ .

$\frac{x + 6 + 3 x^{2} + 9 x - x^{2} - 5 x + 3 x + 3 \cdot 5}{(x + 3) (x + 5)}$

Multiply $3$ by $5$ .

$\frac{x + 6 + 3 x^{2} + 9 x - x^{2} - 5 x + 3 x + 15}{(x + 3) (x + 5)}$

Add $- 5 x$ and $3 x$ .

$\frac{x + 6 + 3 x^{2} + 9 x - x^{2} - 2 x + 15}{(x + 3) (x + 5)}$

Step 5

Simplify by adding terms.

Add $x$ and $9 x$ .

$\frac{10 x + 6 + 3 x^{2} - x^{2} - 2 x + 15}{(x + 3) (x + 5)}$

Subtract $2 x$ from $10 x$ .

$\frac{6 + 3 x^{2} - x^{2} + 8 x + 15}{(x + 3) (x + 5)}$

Add $6$ and $15$ .

$\frac{3 x^{2} - x^{2} + 8 x + 21}{(x + 3) (x + 5)}$

Subtract $x^{2}$ from $3 x^{2}$ .

$\frac{2 x^{2} + 8 x + 21}{(x + 3) (x + 5)}$

Simplify (x+6)/(x^2+8x+15)+(3x)/(x+5)-(x-3)/(x+3)

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