Simplify 4/(x^2-25)+6/(x^2+6x+5)

$\frac{4}{x^{2} - 25} + \frac{6}{x^{2} + 6 x + 5}$

Step 1

Simplify each term.

Simplify the denominator.

Rewrite $25$ as $5^{2}$ .

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

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

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

Factor $x^{2} + 6 x + 5$ 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 $5$ and whose sum is $6$ .

$1, 5$

Write the factored form using these integers.

$\frac{4}{(x + 5) (x - 5)} + \frac{6}{(x + 1) (x + 5)}$

Step 2

To write $\frac{4}{(x + 5) (x - 5)}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

$\frac{4}{(x + 5) (x - 5)} \cdot \frac{x + 1}{x + 1} + \frac{6}{(x + 1) (x + 5)}$

Step 3

To write $\frac{6}{(x + 1) (x + 5)}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

$\frac{4}{(x + 5) (x - 5)} \cdot \frac{x + 1}{x + 1} + \frac{6}{(x + 1) (x + 5)} \cdot \frac{x - 5}{x - 5}$

Step 4

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

Multiply $\frac{4}{(x + 5) (x - 5)}$ by $\frac{x + 1}{x + 1}$ .

$\frac{4 (x + 1)}{(x + 5) (x - 5) (x + 1)} + \frac{6}{(x + 1) (x + 5)} \cdot \frac{x - 5}{x - 5}$

Multiply $\frac{6}{(x + 1) (x + 5)}$ by $\frac{x - 5}{x - 5}$ .

$\frac{4 (x + 1)}{(x + 5) (x - 5) (x + 1)} + \frac{6 (x - 5)}{(x + 1) (x + 5) (x - 5)}$

Reorder the factors of $(x + 5) (x - 5) (x + 1)$ .

$\frac{4 (x + 1)}{(x + 1) (x + 5) (x - 5)} + \frac{6 (x - 5)}{(x + 1) (x + 5) (x - 5)}$

Step 5

Combine the numerators over the common denominator.

$\frac{4 (x + 1) + 6 (x - 5)}{(x + 1) (x + 5) (x - 5)}$

Step 6

Simplify the numerator.

Factor $2$ out of $4 (x + 1) + 6 (x - 5)$ .

Factor $2$ out of $4 (x + 1)$ .

$\frac{2 (2 (x + 1)) + 6 (x - 5)}{(x + 1) (x + 5) (x - 5)}$

Factor $2$ out of $6 (x - 5)$ .

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

Factor $2$ out of $2 (2 (x + 1)) + 2 (3 (x - 5))$ .

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

Apply the distributive property.

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

Multiply $2$ by $1$ .

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

Apply the distributive property.

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

Multiply $3$ by $- 5$ .

$\frac{2 (2 x + 2 + 3 x - 15)}{(x + 1) (x + 5) (x - 5)}$

Add $2 x$ and $3 x$ .

$\frac{2 (5 x + 2 - 15)}{(x + 1) (x + 5) (x - 5)}$

Subtract $15$ from $2$ .

$\frac{2 (5 x - 13)}{(x + 1) (x + 5) (x - 5)}$

Simplify 4/(x^2-25)+6/(x^2+6x+5)

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