Algebra Examples

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

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

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

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Step 2.1

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

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

Step 2.2

Factor $2 x$ out of $- 2 x$ .

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

Step 2.3

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

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

Step 3

Factor $x^{2} + 5 x + 4$ using the AC method.

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Step 3.1

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 $4$ and whose sum is $5$ .

$1, 4$

Step 3.2

Write the factored form using these integers.

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

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $2 x$ .

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Step 4.1.1

Cancel the common factor.

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

Step 4.1.2

Rewrite the expression.

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

Step 4.2

Multiply $\frac{2 x - 1}{(x + 1) (x + 4)}$ by $x^{2} + 2 x + 1$ .

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

Step 5

Factor using the perfect square rule.

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Step 5.1

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

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

Step 5.2

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 5.3

Rewrite the polynomial.

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

Step 5.4

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

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Cancel the common factors.

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Step 6.2.1

Cancel the common factor.

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

Step 6.2.2

Rewrite the expression.

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