Basic Math Examples

$\frac{\frac{\frac{- 3}{m + 2} + 5}{2}}{- (m^{2} + m - 2)}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{\frac{- 3}{m + 2} + 5}{2} \cdot \frac{1}{- (m^{2} + m - 2)}$

Step 2

Simplify the numerator.

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Step 2.1

To write $5$ as a fraction with a common denominator, multiply by $\frac{m + 2}{m + 2}$ .

$\frac{\frac{- 3}{m + 2} + \frac{5 (m + 2)}{m + 2}}{2} \cdot \frac{1}{- (m^{2} + m - 2)}$

Step 2.2

Combine the numerators over the common denominator.

$\frac{\frac{- 3 + 5 (m + 2)}{m + 2}}{2} \cdot \frac{1}{- (m^{2} + m - 2)}$

Step 2.3

Reorder terms.

$\frac{\frac{5 (m + 2) - 3}{m + 2}}{2} \cdot \frac{1}{- (m^{2} + m - 2)}$

Step 2.4

Rewrite $\frac{5 (m + 2) - 3}{m + 2}$ in a factored form.

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Step 2.4.1

Apply the distributive property.

$\frac{\frac{5 m + 5 \cdot 2 - 3}{m + 2}}{2} \cdot \frac{1}{- (m^{2} + m - 2)}$

Step 2.4.2

Multiply $5$ by $2$ .

$\frac{\frac{5 m + 10 - 3}{m + 2}}{2} \cdot \frac{1}{- (m^{2} + m - 2)}$

Step 2.4.3

Subtract $3$ from $10$ .

$\frac{\frac{5 m + 7}{m + 2}}{2} \cdot \frac{1}{- (m^{2} + m - 2)}$

Step 3

Simplify terms.

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Step 3.1

Combine.

$\frac{\frac{5 m + 7}{m + 2} \cdot 1}{2 (- (m^{2} + m - 2))}$

Step 3.2

Cancel the common factor of $1$ and $- 1$ .

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Step 3.2.1

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{\frac{5 m + 7}{m + 2} \cdot (- 1 (- 1))}{2 (- (m^{2} + m - 2))}$

Step 3.2.2

Move the negative in front of the fraction.

$- \frac{\frac{5 m + 7}{m + 2}}{2 (m^{2} + m - 2)}$

Step 4

Multiply the numerator by the reciprocal of the denominator.

$- (\frac{5 m + 7}{m + 2} \cdot \frac{1}{2 (m^{2} + m - 2)})$

Step 5

Factor $m^{2} + m - 2$ using the AC method.

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Step 5.1

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 $- 2$ and whose sum is $1$ .

$- 1, 2$

Step 5.2

Write the factored form using these integers.

$- (\frac{5 m + 7}{m + 2} \cdot \frac{1}{2 ((m - 1) (m + 2))})$

$- (\frac{5 m + 7}{m + 2} \cdot \frac{1}{2 (m - 1) (m + 2)})$

Step 6

Multiply $\frac{5 m + 7}{m + 2} \cdot \frac{1}{2 (m - 1) (m + 2)}$ .

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Step 6.1

Multiply $\frac{5 m + 7}{m + 2}$ by $\frac{1}{2 (m - 1) (m + 2)}$ .

$- \frac{5 m + 7}{(m + 2) (2 (m - 1) (m + 2))}$

Step 6.2

Raise $m + 2$ to the power of $1$ .

$- \frac{5 m + 7}{2 (m - 1) ({(m + 2)}^{1} (m + 2))}$

Step 6.3

Raise $m + 2$ to the power of $1$ .

$- \frac{5 m + 7}{2 (m - 1) ({(m + 2)}^{1} {(m + 2)}^{1})}$

Step 6.4

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

$- \frac{5 m + 7}{2 (m - 1) {(m + 2)}^{1 + 1}}$

Step 6.5

Add $1$ and $1$ .

$- \frac{5 m + 7}{2 (m - 1) {(m + 2)}^{2}}$