Basic Math Examples

$(\frac{m^{2}}{m^{2} - 4} - \frac{m + 2}{m - 2}) \div \frac{4 m + 4}{2 - m}$

Step 1

To divide by a fraction, multiply by its reciprocal.

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

Step 2

Factor $4$ out of $4 m + 4$ .

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Step 2.1

Factor $4$ out of $4 m$ .

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

Step 2.2

Factor $4$ out of $4$ .

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

Step 2.3

Factor $4$ out of $4 (m) + 4 (1)$ .

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

Step 3

Simplify the denominator.

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Step 3.1

Rewrite $4$ as $2^{2}$ .

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

Step 3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = m$ and $b = 2$ .

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

Step 4

To write $- \frac{m + 2}{m - 2}$ as a fraction with a common denominator, multiply by $\frac{m + 2}{m + 2}$ .

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

Step 5

Write each expression with a common denominator of $(m + 2) (m - 2)$ , by multiplying each by an appropriate factor of $1$ .

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Step 5.1

Multiply $\frac{m + 2}{m - 2}$ by $\frac{m + 2}{m + 2}$ .

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

Step 5.2

Reorder the factors of $(m - 2) (m + 2)$ .

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

Step 6

Combine the numerators over the common denominator.

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

Step 7

Simplify the numerator.

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Step 7.1

Rewrite $(m + 2) (m + 2)$ as ${(m + 2)}^{2}$ .

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

Step 7.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = m$ and $b = m + 2$ .

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

Step 7.3

Simplify.

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Step 7.3.1

Add $m$ and $m$ .

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

Step 7.3.2

Factor $2$ out of $2 m + 2$ .

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Step 7.3.2.1

Factor $2$ out of $2 m$ .

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

Step 7.3.2.2

Factor $2$ out of $2$ .

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

Step 7.3.2.3

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

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

Step 7.3.3

Apply the distributive property.

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

Step 7.3.4

Multiply $- 1$ by $2$ .

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

Step 7.3.5

Subtract $m$ from $m$ .

$\frac{2 (m + 1) (0 - 2)}{(m + 2) (m - 2)} \cdot \frac{2 - m}{4 (m + 1)}$

Step 7.3.6

Subtract $2$ from $0$ .

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

Step 7.3.7

Multiply $- 2$ by $2$ .

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

Step 8

Simplify terms.

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Step 8.1

Cancel the common factor of $4 (m + 1)$ .

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Step 8.1.1

Factor $4 (m + 1)$ out of $- 4 (m + 1)$ .

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

Step 8.1.2

Cancel the common factor.

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

Step 8.1.3

Rewrite the expression.

$\frac{- 1}{(m + 2) (m - 2)} (2 - m)$

Step 8.2

Move the negative in front of the fraction.

$- \frac{1}{(m + 2) (m - 2)} (2 - m)$

Step 8.3

Apply the distributive property.

$- \frac{1}{(m + 2) (m - 2)} \cdot 2 - \frac{1}{(m + 2) (m - 2)} (- m)$

Step 9

Multiply $- \frac{1}{(m + 2) (m - 2)} \cdot 2$ .

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Step 9.1

Multiply $2$ by $- 1$ .

$- 2 \frac{1}{(m + 2) (m - 2)} - \frac{1}{(m + 2) (m - 2)} (- m)$

Step 9.2

Combine $- 2$ and $\frac{1}{(m + 2) (m - 2)}$ .

$\frac{- 2}{(m + 2) (m - 2)} - \frac{1}{(m + 2) (m - 2)} (- m)$

Step 10

Multiply $- \frac{1}{(m + 2) (m - 2)} (- m)$ .

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Step 10.1

Multiply $- 1$ by $- 1$ .

$\frac{- 2}{(m + 2) (m - 2)} + 1 \frac{1}{(m + 2) (m - 2)} m$

Step 10.2

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

$\frac{- 2}{(m + 2) (m - 2)} + \frac{1}{(m + 2) (m - 2)} m$

Step 10.3

Combine $\frac{1}{(m + 2) (m - 2)}$ and $m$ .

$\frac{- 2}{(m + 2) (m - 2)} + \frac{m}{(m + 2) (m - 2)}$

Step 11

Combine the numerators over the common denominator.

$\frac{- 2 + m}{(m + 2) (m - 2)}$

Step 12

Cancel the common factor of $- 2 + m$ and $m - 2$ .

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Step 12.1

Reorder terms.

$\frac{m - 2}{(m + 2) (m - 2)}$

Step 12.2

Cancel the common factor.

$\frac{m - 2}{(m + 2) (m - 2)}$

Step 12.3

Rewrite the expression.

$\frac{1}{m + 2}$