Algebra Examples

$\frac{- 8 m^{2} + 8 m - 2}{32 m^{2} - 4 m - 6}$

Step 1

Cancel the common factor of $- 8 m^{2} + 8 m - 2$ and $32 m^{2} - 4 m - 6$ .

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Step 1.1

Factor $2$ out of $- 8 m^{2}$ .

$\frac{2 (- 4 m^{2}) + 8 m - 2}{32 m^{2} - 4 m - 6}$

Step 1.2

Factor $2$ out of $8 m$ .

$\frac{2 (- 4 m^{2}) + 2 (4 m) - 2}{32 m^{2} - 4 m - 6}$

Step 1.3

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

$\frac{2 (- 4 m^{2} + 4 m) - 2}{32 m^{2} - 4 m - 6}$

Step 1.4

Factor $2$ out of $- 2$ .

$\frac{2 (- 4 m^{2} + 4 m) + 2 (- 1)}{32 m^{2} - 4 m - 6}$

Step 1.5

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

$\frac{2 (- 4 m^{2} + 4 m - 1)}{32 m^{2} - 4 m - 6}$

Step 1.6

Cancel the common factors.

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Step 1.6.1

Factor $2$ out of $32 m^{2}$ .

$\frac{2 (- 4 m^{2} + 4 m - 1)}{2 (16 m^{2}) - 4 m - 6}$

Step 1.6.2

Factor $2$ out of $- 4 m$ .

$\frac{2 (- 4 m^{2} + 4 m - 1)}{2 (16 m^{2}) + 2 (- 2 m) - 6}$

Step 1.6.3

Factor $2$ out of $2 (16 m^{2}) + 2 (- 2 m)$ .

$\frac{2 (- 4 m^{2} + 4 m - 1)}{2 (16 m^{2} - 2 m) - 6}$

Step 1.6.4

Factor $2$ out of $- 6$ .

$\frac{2 (- 4 m^{2} + 4 m - 1)}{2 (16 m^{2} - 2 m) + 2 (- 3)}$

Step 1.6.5

Factor $2$ out of $2 (16 m^{2} - 2 m) + 2 (- 3)$ .

$\frac{2 (- 4 m^{2} + 4 m - 1)}{2 (16 m^{2} - 2 m - 3)}$

Step 1.6.6

Cancel the common factor.

$\frac{2 (- 4 m^{2} + 4 m - 1)}{2 (16 m^{2} - 2 m - 3)}$

Step 1.6.7

Rewrite the expression.

$\frac{- 4 m^{2} + 4 m - 1}{16 m^{2} - 2 m - 3}$

Step 2

Factor by grouping.

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Step 2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 4 \cdot - 1 = 4$ and whose sum is $b = 4$ .

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Step 2.1.1

Factor $4$ out of $4 m$ .

$\frac{- 4 m^{2} + 4 (m) - 1}{16 m^{2} - 2 m - 3}$

Step 2.1.2

Rewrite $4$ as $2$ plus $2$

$\frac{- 4 m^{2} + (2 + 2) m - 1}{16 m^{2} - 2 m - 3}$

Step 2.1.3

Apply the distributive property.

$\frac{- 4 m^{2} + 2 m + 2 m - 1}{16 m^{2} - 2 m - 3}$

Step 2.2

Factor out the greatest common factor from each group.

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Step 2.2.1

Group the first two terms and the last two terms.

$\frac{(- 4 m^{2} + 2 m) + 2 m - 1}{16 m^{2} - 2 m - 3}$

Step 2.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{2 m (- 2 m + 1) - (- 2 m + 1)}{16 m^{2} - 2 m - 3}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $- 2 m + 1$ .

$\frac{(- 2 m + 1) (2 m - 1)}{16 m^{2} - 2 m - 3}$

Step 3

Factor by grouping.

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Step 3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 16 \cdot - 3 = - 48$ and whose sum is $b = - 2$ .

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Step 3.1.1

Factor $- 2$ out of $- 2 m$ .

$\frac{(- 2 m + 1) (2 m - 1)}{16 m^{2} - 2 (m) - 3}$

Step 3.1.2

Rewrite $- 2$ as $6$ plus $- 8$

$\frac{(- 2 m + 1) (2 m - 1)}{16 m^{2} + (6 - 8) m - 3}$

Step 3.1.3

Apply the distributive property.

$\frac{(- 2 m + 1) (2 m - 1)}{16 m^{2} + 6 m - 8 m - 3}$

Step 3.2

Factor out the greatest common factor from each group.

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Step 3.2.1

Group the first two terms and the last two terms.

$\frac{(- 2 m + 1) (2 m - 1)}{(16 m^{2} + 6 m) - 8 m - 3}$

Step 3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{(- 2 m + 1) (2 m - 1)}{2 m (8 m + 3) - (8 m + 3)}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $8 m + 3$ .

$\frac{(- 2 m + 1) (2 m - 1)}{(8 m + 3) (2 m - 1)}$

Step 4

Simplify the numerator.

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Step 4.1

Factor $- 1$ out of $- 2 m$ .

$\frac{(- (2 m) + 1) (2 m - 1)}{(8 m + 3) (2 m - 1)}$

Step 4.2

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

$\frac{(- (2 m) - 1 (- 1)) (2 m - 1)}{(8 m + 3) (2 m - 1)}$

Step 4.3

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

$\frac{- (2 m - 1) (2 m - 1)}{(8 m + 3) (2 m - 1)}$

Step 4.4

Rewrite $- (2 m - 1)$ as $- 1 (2 m - 1)$ .

$\frac{- 1 (2 m - 1) (2 m - 1)}{(8 m + 3) (2 m - 1)}$

Step 4.5

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

$\frac{- 1 ({(2 m - 1)}^{1} (2 m - 1))}{(8 m + 3) (2 m - 1)}$

Step 4.6

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

$\frac{- 1 ({(2 m - 1)}^{1} {(2 m - 1)}^{1})}{(8 m + 3) (2 m - 1)}$

Step 4.7

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

$\frac{- 1 {(2 m - 1)}^{1 + 1}}{(8 m + 3) (2 m - 1)}$

Step 4.8

Add $1$ and $1$ .

$\frac{- 1 {(2 m - 1)}^{2}}{(8 m + 3) (2 m - 1)}$

Step 5

Reduce the expression by cancelling the common factors.

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Step 5.1

Cancel the common factor of ${(2 m - 1)}^{2}$ and $2 m - 1$ .

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Step 5.1.1

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

$\frac{(2 m - 1) (- 1 (2 m - 1))}{(8 m + 3) (2 m - 1)}$

Step 5.1.2

Cancel the common factors.

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Step 5.1.2.1

Factor $2 m - 1$ out of $(8 m + 3) (2 m - 1)$ .

$\frac{(2 m - 1) (- 1 (2 m - 1))}{(2 m - 1) (8 m + 3)}$

Step 5.1.2.2

Cancel the common factor.

$\frac{(2 m - 1) (- 1 (2 m - 1))}{(2 m - 1) (8 m + 3)}$

Step 5.1.2.3

Rewrite the expression.

$\frac{- 1 (2 m - 1)}{8 m + 3}$

Step 5.2

Move the negative in front of the fraction.

$- \frac{2 m - 1}{8 m + 3}$