Simplify (x+3)/(x+5)+6/(x^2+3x-10)

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

Factor $x^{2} + 3 x - 10$ 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 $- 10$ and whose sum is $3$ .

$- 2, 5$

Write the factored form using these integers.

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

To write $\frac{x + 3}{x + 5}$ as a fraction with a common denominator, multiply by $\frac{x - 2}{x - 2}$ .

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

Write each expression with a common denominator of $(x + 5) (x - 2)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{x + 3}{x + 5}$ by $\frac{x - 2}{x - 2}$ .

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

Reorder the factors of $(x - 2) (x + 5)$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

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

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

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

Move $- 2$ to the left of $x$ .

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

Multiply $3$ by $- 2$ .

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

Add $- 2 x$ and $3 x$ .

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

Add $- 6$ and $6$ .

$\frac{x^{2} + x + 0}{(x + 5) (x - 2)}$

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

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

Factor $x$ out of $x^{2} + x$ .

Factor $x$ out of $x^{2}$ .

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

Raise $x$ to the power of $1$ .

$\frac{x \cdot x + x^{1}}{(x + 5) (x - 2)}$

Factor $x$ out of $x^{1}$ .

$\frac{x \cdot x + x \cdot 1}{(x + 5) (x - 2)}$

Factor $x$ out of $x \cdot x + x \cdot 1$ .

$\frac{x (x + 1)}{(x + 5) (x - 2)}$

Simplify (x+3)/(x+5)+6/(x^2+3x-10)

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