Simplify (4x)/(x^2+9x+18)+5/(x+6)

$\frac{4 x}{x^{2} + 9 x + 18} + \frac{5}{x + 6}$

Step 1

Factor $x^{2} + 9 x + 18$ 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 $18$ and whose sum is $9$ .

$3, 6$

Write the factored form using these integers.

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

Step 2

To write $\frac{5}{x + 6}$ as a fraction with a common denominator, multiply by $\frac{x + 3}{x + 3}$ .

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

Step 3

Write each expression with a common denominator of $(x + 3) (x + 6)$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{5}{x + 6}$ by $\frac{x + 3}{x + 3}$ .

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

Reorder the factors of $(x + 6) (x + 3)$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $5$ by $3$ .

$\frac{4 x + 5 x + 15}{(x + 3) (x + 6)}$

Add $4 x$ and $5 x$ .

$\frac{9 x + 15}{(x + 3) (x + 6)}$

Factor $3$ out of $9 x + 15$ .

Factor $3$ out of $9 x$ .

$\frac{3 (3 x) + 15}{(x + 3) (x + 6)}$

Factor $3$ out of $15$ .

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

Factor $3$ out of $3 (3 x) + 3 (5)$ .

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

Simplify (4x)/(x^2+9x+18)+5/(x+6)

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