Algebra Examples

$\frac{2 x^{2} + 4 x + 2}{x + 1} \div \frac{2}{x}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Simplify the numerator.

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Step 2.1

Factor $2$ out of $2 x^{2} + 4 x + 2$ .

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Step 2.1.1

Factor $2$ out of $2 x^{2}$ .

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

Step 2.1.2

Factor $2$ out of $4 x$ .

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

Step 2.1.3

Factor $2$ out of $2$ .

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

Step 2.1.4

Factor $2$ out of $2 (x^{2}) + 2 (2 x)$ .

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

Step 2.1.5

Factor $2$ out of $2 (x^{2} + 2 x) + 2 (1)$ .

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

Step 2.2

Factor using the perfect square rule.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.2.3

Rewrite the polynomial.

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

Step 2.2.4

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

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

Step 3

Cancel the common factor of $2$ .

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Step 3.1

Factor $2$ out of $2 {(x + 1)}^{2}$ .

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

Step 3.2

Cancel the common factor.

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

Step 3.3

Rewrite the expression.

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

Step 4

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

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

Step 5

Cancel the common factor of ${(x + 1)}^{2}$ and $x + 1$ .

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Step 5.1

Factor $x + 1$ out of ${(x + 1)}^{2} x$ .

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

Step 5.2

Cancel the common factors.

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Step 5.2.1

Multiply by $1$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

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

Step 5.2.4

Divide $(x + 1) x$ by $1$ .

$(x + 1) x$

Step 6

Apply the distributive property.

$x \cdot x + 1 x$

Step 7

Multiply $x$ by $x$ .

$x^{2} + 1 x$

Step 8

Multiply $x$ by $1$ .

$x^{2} + x$