Simplify 1/(x^2+2x+1)+x/(x+1)

$\frac{1}{x^{2} + 2 x + 1} + \frac{x}{x + 1}$

Step 1

Factor using the perfect square rule.

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

$\frac{1}{x^{2} + 2 x + 1^{2}} + \frac{x}{x + 1}$

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.

$\frac{1}{x^{2} + 2 \cdot x \cdot 1 + 1^{2}} + \frac{x}{x + 1}$

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

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

Step 2

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

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

Step 3

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

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

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

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

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

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

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

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

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

Add $1$ and $1$ .

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

Step 4

Combine the numerators over the common denominator.

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

Step 5

Simplify the numerator.

Apply the distributive property.

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

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

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

Reorder terms.

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