Basic Math Examples

$\frac{6 m^{2} + m - 12}{2 m^{2} - 5 m - 12} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

Step 1

Factor by grouping.

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Step 1.1

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 $a \cdot c = 6 \cdot - 12 = - 72$ and whose sum is $b = 1$ .

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Step 1.1.1

Multiply by $1$ .

$\frac{6 m^{2} + 1 m - 12}{2 m^{2} - 5 m - 12} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

Step 1.1.2

Rewrite $1$ as $- 8$ plus $9$

$\frac{6 m^{2} + (- 8 + 9) m - 12}{2 m^{2} - 5 m - 12} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

Step 1.1.3

Apply the distributive property.

$\frac{6 m^{2} - 8 m + 9 m - 12}{2 m^{2} - 5 m - 12} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

Step 1.2

Factor out the greatest common factor from each group.

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Step 1.2.1

Group the first two terms and the last two terms.

$\frac{(6 m^{2} - 8 m) + 9 m - 12}{2 m^{2} - 5 m - 12} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

Step 1.2.2

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

$\frac{2 m (3 m - 4) + 3 (3 m - 4)}{2 m^{2} - 5 m - 12} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $3 m - 4$ .

$\frac{(3 m - 4) (2 m + 3)}{2 m^{2} - 5 m - 12} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

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 $a \cdot c = 2 \cdot - 12 = - 24$ and whose sum is $b = - 5$ .

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Step 2.1.1

Factor $- 5$ out of $- 5 m$ .

$\frac{(3 m - 4) (2 m + 3)}{2 m^{2} - 5 (m) - 12} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

Step 2.1.2

Rewrite $- 5$ as $3$ plus $- 8$

$\frac{(3 m - 4) (2 m + 3)}{2 m^{2} + (3 - 8) m - 12} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

Step 2.1.3

Apply the distributive property.

$\frac{(3 m - 4) (2 m + 3)}{2 m^{2} + 3 m - 8 m - 12} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

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.

$\frac{(3 m - 4) (2 m + 3)}{(2 m^{2} + 3 m) - 8 m - 12} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

Step 2.2.2

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

$\frac{(3 m - 4) (2 m + 3)}{m (2 m + 3) - 4 (2 m + 3)} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

Step 2.3

Factor the polynomial by factoring out the greatest common factor, $2 m + 3$ .

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{3 m^{2} - 14 m + 8}{9 m^{2} - 18 m + 8}$

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 $a \cdot c = 3 \cdot 8 = 24$ and whose sum is $b = - 14$ .

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Step 3.1.1

Factor $- 14$ out of $- 14 m$ .

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{3 m^{2} - 14 (m) + 8}{9 m^{2} - 18 m + 8}$

Step 3.1.2

Rewrite $- 14$ as $- 2$ plus $- 12$

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{3 m^{2} + (- 2 - 12) m + 8}{9 m^{2} - 18 m + 8}$

Step 3.1.3

Apply the distributive property.

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{3 m^{2} - 2 m - 12 m + 8}{9 m^{2} - 18 m + 8}$

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.

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{(3 m^{2} - 2 m) - 12 m + 8}{9 m^{2} - 18 m + 8}$

Step 3.2.2

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

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{m (3 m - 2) - 4 (3 m - 2)}{9 m^{2} - 18 m + 8}$

Step 3.3

Factor the polynomial by factoring out the greatest common factor, $3 m - 2$ .

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{(3 m - 2) (m - 4)}{9 m^{2} - 18 m + 8}$

Step 4

Factor by grouping.

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Step 4.1

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 $a \cdot c = 9 \cdot 8 = 72$ and whose sum is $b = - 18$ .

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Step 4.1.1

Factor $- 18$ out of $- 18 m$ .

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{(3 m - 2) (m - 4)}{9 m^{2} - 18 (m) + 8}$

Step 4.1.2

Rewrite $- 18$ as $- 6$ plus $- 12$

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{(3 m - 2) (m - 4)}{9 m^{2} + (- 6 - 12) m + 8}$

Step 4.1.3

Apply the distributive property.

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{(3 m - 2) (m - 4)}{9 m^{2} - 6 m - 12 m + 8}$

Step 4.2

Factor out the greatest common factor from each group.

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Step 4.2.1

Group the first two terms and the last two terms.

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{(3 m - 2) (m - 4)}{(9 m^{2} - 6 m) - 12 m + 8}$

Step 4.2.2

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

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{(3 m - 2) (m - 4)}{3 m (3 m - 2) - 4 (3 m - 2)}$

Step 4.3

Factor the polynomial by factoring out the greatest common factor, $3 m - 2$ .

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{(3 m - 2) (m - 4)}{(3 m - 2) (3 m - 4)}$

Step 5

Cancel the common factor of $3 m - 4$ .

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Step 5.1

Factor $3 m - 4$ out of $(3 m - 2) (3 m - 4)$ .

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{(3 m - 2) (m - 4)}{(3 m - 4) (3 m - 2)}$

Step 5.2

Cancel the common factor.

$\frac{(3 m - 4) (2 m + 3)}{(2 m + 3) (m - 4)} \cdot \frac{(3 m - 2) (m - 4)}{(3 m - 4) (3 m - 2)}$

Step 5.3

Rewrite the expression.

$\frac{2 m + 3}{(2 m + 3) (m - 4)} \cdot \frac{(3 m - 2) (m - 4)}{3 m - 2}$

Step 6

Cancel the common factor of $m - 4$ .

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Step 6.1

Factor $m - 4$ out of $(2 m + 3) (m - 4)$ .

$\frac{2 m + 3}{(m - 4) (2 m + 3)} \cdot \frac{(3 m - 2) (m - 4)}{3 m - 2}$

Step 6.2

Factor $m - 4$ out of $(3 m - 2) (m - 4)$ .

$\frac{2 m + 3}{(m - 4) (2 m + 3)} \cdot \frac{(m - 4) (3 m - 2)}{3 m - 2}$

Step 6.3

Cancel the common factor.

$\frac{2 m + 3}{(m - 4) (2 m + 3)} \cdot \frac{(m - 4) (3 m - 2)}{3 m - 2}$

Step 6.4

Rewrite the expression.

$\frac{2 m + 3}{2 m + 3} \cdot \frac{3 m - 2}{3 m - 2}$

Step 7

Multiply $\frac{2 m + 3}{2 m + 3}$ by $\frac{3 m - 2}{3 m - 2}$ .

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

Step 8

Cancel the common factor of $2 m + 3$ .

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Step 8.1

Cancel the common factor.

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

Step 8.2

Rewrite the expression.

$\frac{3 m - 2}{3 m - 2}$

Step 9

Cancel the common factor of $3 m - 2$ .

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Step 9.1

Cancel the common factor.

$\frac{3 m - 2}{3 m - 2}$

Step 9.2

Rewrite the expression.

$1$