Simplify (5x)/(x^2-x-6)+4/(x^2+4x+4)

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

Step 1

Simplify each term.

Factor $x^{2} - x - 6$ 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 $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Write the factored form using these integers.

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

Factor using the perfect square rule.

Rewrite $4$ as $2^{2}$ .

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

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot x \cdot 2$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 2$ .

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

Step 2

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

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

Step 3

To write $\frac{4}{{(x + 2)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x - 3}{x - 3}$ .

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

Step 4

Write each expression with a common denominator of $(x - 3) {(x + 2)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

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

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

Raise $x + 2$ to the power of $1$ .

$\frac{5 x (x + 2)}{(x - 3) ({(x + 2)}^{1} (x + 2))} + \frac{4}{{(x + 2)}^{2}} \cdot \frac{x - 3}{x - 3}$

Raise $x + 2$ to the power of $1$ .

$\frac{5 x (x + 2)}{(x - 3) ({(x + 2)}^{1} {(x + 2)}^{1})} + \frac{4}{{(x + 2)}^{2}} \cdot \frac{x - 3}{x - 3}$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{5 x (x + 2)}{(x - 3) {(x + 2)}^{1 + 1}} + \frac{4}{{(x + 2)}^{2}} \cdot \frac{x - 3}{x - 3}$

Add $1$ and $1$ .

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

Multiply $\frac{4}{{(x + 2)}^{2}}$ by $\frac{x - 3}{x - 3}$ .

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

Reorder the factors of $(x - 3) {(x + 2)}^{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $2$ by $5$ .

$\frac{5 x^{2} + 10 x + 4 (x - 3)}{{(x + 2)}^{2} (x - 3)}$

Apply the distributive property.

$\frac{5 x^{2} + 10 x + 4 x + 4 \cdot - 3}{{(x + 2)}^{2} (x - 3)}$

Multiply $4$ by $- 3$ .

$\frac{5 x^{2} + 10 x + 4 x - 12}{{(x + 2)}^{2} (x - 3)}$

Add $10 x$ and $4 x$ .

$\frac{5 x^{2} + 14 x - 12}{{(x + 2)}^{2} (x - 3)}$

Simplify (5x)/(x^2-x-6)+4/(x^2+4x+4)

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