Simplify 4/(x+2)-(3x+18)/(x^2+7x+10)

$\frac{4}{x + 2} - \frac{3 x + 18}{x^{2} + 7 x + 10}$

Simplify each term.

Factor $3$ out of $3 x + 18$ .

Factor $3$ out of $3 x$ .

$\frac{4}{x + 2} - \frac{3 (x) + 18}{x^{2} + 7 x + 10}$

Factor $3$ out of $18$ .

$\frac{4}{x + 2} - \frac{3 x + 3 \cdot 6}{x^{2} + 7 x + 10}$

Factor $3$ out of $3 x + 3 \cdot 6$ .

$\frac{4}{x + 2} - \frac{3 (x + 6)}{x^{2} + 7 x + 10}$

Factor $x^{2} + 7 x + 10$ 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 $7$ .

$2, 5$

Write the factored form using these integers.

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

To write $\frac{4}{x + 2}$ as a fraction with a common denominator, multiply by $\frac{x + 5}{x + 5}$ .

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

Simplify terms.

Multiply $\frac{4}{x + 2}$ by $\frac{x + 5}{x + 5}$ .

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

Combine the numerators over the common denominator.

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

Simplify the numerator.

Apply the distributive property.

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

Multiply $4$ by $5$ .

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

Apply the distributive property.

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

Multiply $- 3$ by $6$ .

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

Subtract $3 x$ from $4 x$ .

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

Subtract $18$ from $20$ .

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

Cancel the common factor of $x + 2$ .

Cancel the common factor.

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

Rewrite the expression.

$\frac{1}{x + 5}$

Simplify 4/(x+2)-(3x+18)/(x^2+7x+10)

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