Simplify (6x-7)/(x^2+6x+5)+4/(x+5)

$\frac{6 x - 7}{x^{2} + 6 x + 5} + \frac{4}{x + 5}$

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.

$\frac{6 x - 7}{(x + 1) (x + 5)} + \frac{4}{x + 5}$

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

$\frac{6 x - 7}{(x + 1) (x + 5)} + \frac{4}{x + 5} \cdot \frac{x + 1}{x + 1}$

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

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

$\frac{6 x - 7}{(x + 1) (x + 5)} + \frac{4 (x + 1)}{(x + 5) (x + 1)}$

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

$\frac{6 x - 7}{(x + 1) (x + 5)} + \frac{4 (x + 1)}{(x + 1) (x + 5)}$

Combine the numerators over the common denominator.

$\frac{6 x - 7 + 4 (x + 1)}{(x + 1) (x + 5)}$

Simplify the numerator.

Apply the distributive property.

$\frac{6 x - 7 + 4 x + 4 \cdot 1}{(x + 1) (x + 5)}$

Multiply $4$ by $1$ .

$\frac{6 x - 7 + 4 x + 4}{(x + 1) (x + 5)}$

Add $6 x$ and $4 x$ .

$\frac{10 x - 7 + 4}{(x + 1) (x + 5)}$

Add $- 7$ and $4$ .

$\frac{10 x - 3}{(x + 1) (x + 5)}$

Simplify (6x-7)/(x^2+6x+5)+4/(x+5)

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