Algebra Examples

$\frac{2 r + 2}{r + 2} \div \frac{4 r^{2} + 8 r + 4}{12 r + 12}$

Step 1

To divide by a fraction, multiply by its reciprocal.

$\frac{2 r + 2}{r + 2} \cdot \frac{12 r + 12}{4 r^{2} + 8 r + 4}$

Step 2

Simplify terms.

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Step 2.1

Factor $2$ out of $2 r + 2$ .

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Step 2.1.1

Factor $2$ out of $2 r$ .

$\frac{2 (r) + 2}{r + 2} \cdot \frac{12 r + 12}{4 r^{2} + 8 r + 4}$

Step 2.1.2

Factor $2$ out of $2$ .

$\frac{2 (r) + 2 (1)}{r + 2} \cdot \frac{12 r + 12}{4 r^{2} + 8 r + 4}$

Step 2.1.3

Factor $2$ out of $2 (r) + 2 (1)$ .

$\frac{2 (r + 1)}{r + 2} \cdot \frac{12 r + 12}{4 r^{2} + 8 r + 4}$

Step 2.2

Cancel the common factor of $12 r + 12$ and $4 r^{2} + 8 r + 4$ .

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Step 2.2.1

Factor $4$ out of $12 r$ .

$\frac{2 (r + 1)}{r + 2} \cdot \frac{4 (3 r) + 12}{4 r^{2} + 8 r + 4}$

Step 2.2.2

Factor $4$ out of $12$ .

$\frac{2 (r + 1)}{r + 2} \cdot \frac{4 (3 r) + 4 \cdot 3}{4 r^{2} + 8 r + 4}$

Step 2.2.3

Factor $4$ out of $4 (3 r) + 4 (3)$ .

$\frac{2 (r + 1)}{r + 2} \cdot \frac{4 (3 r + 3)}{4 r^{2} + 8 r + 4}$

Step 2.2.4

Cancel the common factors.

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Step 2.2.4.1

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

$\frac{2 (r + 1)}{r + 2} \cdot \frac{4 (3 r + 3)}{4 (r^{2}) + 8 r + 4}$

Step 2.2.4.2

Factor $4$ out of $8 r$ .

$\frac{2 (r + 1)}{r + 2} \cdot \frac{4 (3 r + 3)}{4 (r^{2}) + 4 (2 r) + 4}$

Step 2.2.4.3

Factor $4$ out of $4 (r^{2}) + 4 (2 r)$ .

$\frac{2 (r + 1)}{r + 2} \cdot \frac{4 (3 r + 3)}{4 (r^{2} + 2 r) + 4}$

Step 2.2.4.4

Factor $4$ out of $4$ .

$\frac{2 (r + 1)}{r + 2} \cdot \frac{4 (3 r + 3)}{4 (r^{2} + 2 r) + 4 (1)}$

Step 2.2.4.5

Factor $4$ out of $4 (r^{2} + 2 r) + 4 (1)$ .

$\frac{2 (r + 1)}{r + 2} \cdot \frac{4 (3 r + 3)}{4 (r^{2} + 2 r + 1)}$

Step 2.2.4.6

Cancel the common factor.

$\frac{2 (r + 1)}{r + 2} \cdot \frac{4 (3 r + 3)}{4 (r^{2} + 2 r + 1)}$

Step 2.2.4.7

Rewrite the expression.

$\frac{2 (r + 1)}{r + 2} \cdot \frac{3 r + 3}{r^{2} + 2 r + 1}$

Step 2.3

Factor $3$ out of $3 r + 3$ .

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Step 2.3.1

Factor $3$ out of $3 r$ .

$\frac{2 (r + 1)}{r + 2} \cdot \frac{3 (r) + 3}{r^{2} + 2 r + 1}$

Step 2.3.2

Factor $3$ out of $3$ .

$\frac{2 (r + 1)}{r + 2} \cdot \frac{3 (r) + 3 (1)}{r^{2} + 2 r + 1}$

Step 2.3.3

Factor $3$ out of $3 (r) + 3 (1)$ .

$\frac{2 (r + 1)}{r + 2} \cdot \frac{3 (r + 1)}{r^{2} + 2 r + 1}$

Step 3

Factor using the perfect square rule.

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Step 3.1

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

$\frac{2 (r + 1)}{r + 2} \cdot \frac{3 (r + 1)}{r^{2} + 2 r + 1^{2}}$

Step 3.2

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

$2 r = 2 \cdot r \cdot 1$

Step 3.3

Rewrite the polynomial.

$\frac{2 (r + 1)}{r + 2} \cdot \frac{3 (r + 1)}{r^{2} + 2 \cdot r \cdot 1 + 1^{2}}$

Step 3.4

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

$\frac{2 (r + 1)}{r + 2} \cdot \frac{3 (r + 1)}{{(r + 1)}^{2}}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $r + 1$ .

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Step 4.1.1

Factor $r + 1$ out of $2 (r + 1)$ .

$\frac{(r + 1) \cdot 2}{r + 2} \cdot \frac{3 (r + 1)}{{(r + 1)}^{2}}$

Step 4.1.2

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

$\frac{(r + 1) \cdot 2}{r + 2} \cdot \frac{3 (r + 1)}{(r + 1) (r + 1)}$

Step 4.1.3

Cancel the common factor.

$\frac{(r + 1) \cdot 2}{r + 2} \cdot \frac{3 (r + 1)}{(r + 1) (r + 1)}$

Step 4.1.4

Rewrite the expression.

$\frac{2}{r + 2} \cdot \frac{3 (r + 1)}{r + 1}$

Step 4.2

Multiply $\frac{2}{r + 2}$ by $\frac{3 (r + 1)}{r + 1}$ .

$\frac{2 (3 (r + 1))}{(r + 2) (r + 1)}$

Step 4.3

Multiply $3$ by $2$ .

$\frac{6 (r + 1)}{(r + 2) (r + 1)}$

Step 4.4

Cancel the common factor of $r + 1$ .

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Step 4.4.1

Cancel the common factor.

$\frac{6 (r + 1)}{(r + 2) (r + 1)}$

Step 4.4.2

Rewrite the expression.

$\frac{6}{r + 2}$