Simplify (3x)/(x^2+2x+1)+2/(x^2-4x-5)

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

Step 1

Simplify each term.

Factor using the perfect square rule.

Rewrite $1$ as $1^{2}$ .

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

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

$2 x = 2 \cdot x \cdot 1$

Rewrite the polynomial.

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

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Factor $x^{2} - 4 x - 5$ 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 $- 4$ .

$- 5, 1$

Write the factored form using these integers.

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

Step 2

To write $\frac{3 x}{{(x + 1)}^{2}}$ as a fraction with a common denominator, multiply by $\frac{x - 5}{x - 5}$ .

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

Step 3

To write $\frac{2}{(x - 5) (x + 1)}$ as a fraction with a common denominator, multiply by $\frac{x + 1}{x + 1}$ .

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

Step 4

Write each expression with a common denominator of $(x - 5) {(x + 1)}^{2}$ , by multiplying each by an appropriate factor of $1$ .

Multiply $\frac{3 x}{{(x + 1)}^{2}}$ by $\frac{x - 5}{x - 5}$ .

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

Multiply $\frac{2}{(x - 5) (x + 1)}$ by $\frac{x + 1}{x + 1}$ .

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Reorder the factors of $(x - 5) {(x + 1)}^{2}$ .

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

Step 5

Combine the numerators over the common denominator.

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

Step 6

Simplify the numerator.

Apply the distributive property.

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $- 5$ by $3$ .

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

Apply the distributive property.

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

Multiply $2$ by $1$ .

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

Add $- 15 x$ and $2 x$ .

$\frac{3 x^{2} - 13 x + 2}{{(x + 1)}^{2} (x - 5)}$

Simplify (3x)/(x^2+2x+1)+2/(x^2-4x-5)

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