Finite Math Examples

$\frac{x - 5}{4 x + 8} \cdot (12 x^{2} + 32 x + 16)$

Step 1

Factor $4$ out of $4 x + 8$ .

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Step 1.1

Factor $4$ out of $4 x$ .

$\frac{x - 5}{4 (x) + 8} \cdot (12 x^{2} + 32 x + 16)$

Step 1.2

Factor $4$ out of $8$ .

$\frac{x - 5}{4 x + 4 \cdot 2} \cdot (12 x^{2} + 32 x + 16)$

Step 1.3

Factor $4$ out of $4 x + 4 \cdot 2$ .

$\frac{x - 5}{4 (x + 2)} \cdot (12 x^{2} + 32 x + 16)$

Step 2

Multiply $\frac{x - 5}{4 (x + 2)}$ by $12 x^{2} + 32 x + 16$ .

$\frac{(x - 5) (12 x^{2} + 32 x + 16)}{4 (x + 2)}$

Step 3

Cancel the common factor of $12 x^{2} + 32 x + 16$ and $4$ .

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Step 3.1

Factor $4$ out of $(x - 5) (12 x^{2} + 32 x + 16)$ .

$\frac{4 ((x - 5) (3 x^{2} + 8 x + 4))}{4 (x + 2)}$

Step 3.2

Cancel the common factors.

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Step 3.2.1

Cancel the common factor.

$\frac{4 ((x - 5) (3 x^{2} + 8 x + 4))}{4 (x + 2)}$

Step 3.2.2

Rewrite the expression.

$\frac{(x - 5) (3 x^{2} + 8 x + 4)}{x + 2}$

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 $a \cdot c = 3 \cdot 4 = 12$ and whose sum is $b = 8$ .

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Step 4.1.1

Factor $8$ out of $8 x$ .

$\frac{(x - 5) (3 x^{2} + 8 (x) + 4)}{x + 2}$

Step 4.1.2

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

$\frac{(x - 5) (3 x^{2} + (2 + 6) x + 4)}{x + 2}$

Step 4.1.3

Apply the distributive property.

$\frac{(x - 5) (3 x^{2} + 2 x + 6 x + 4)}{x + 2}$

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.

$\frac{(x - 5) ((3 x^{2} + 2 x) + 6 x + 4)}{x + 2}$

Step 4.2.2

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

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

Step 4.3

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

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

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

Step 5

Cancel the common factor of $x + 2$ .

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Step 5.1

Cancel the common factor.

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

Step 5.2

Divide $(x - 5) (3 x + 2)$ by $1$ .

$(x - 5) (3 x + 2)$

Step 6

Expand $(x - 5) (3 x + 2)$ using the FOIL Method.

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Step 6.1

Apply the distributive property.

$x (3 x + 2) - 5 (3 x + 2)$

Step 6.2

Apply the distributive property.

$x (3 x) + x \cdot 2 - 5 (3 x + 2)$

Step 6.3

Apply the distributive property.

$x (3 x) + x \cdot 2 - 5 (3 x) - 5 \cdot 2$

Step 7

Simplify and combine like terms.

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Step 7.1

Simplify each term.

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Step 7.1.1

Rewrite using the commutative property of multiplication.

$3 x \cdot x + x \cdot 2 - 5 (3 x) - 5 \cdot 2$

Step 7.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 7.1.2.1

Move $x$ .

$3 (x \cdot x) + x \cdot 2 - 5 (3 x) - 5 \cdot 2$

Step 7.1.2.2

Multiply $x$ by $x$ .

$3 x^{2} + x \cdot 2 - 5 (3 x) - 5 \cdot 2$

Step 7.1.3

Move $2$ to the left of $x$ .

$3 x^{2} + 2 \cdot x - 5 (3 x) - 5 \cdot 2$

Step 7.1.4

Multiply $3$ by $- 5$ .

$3 x^{2} + 2 x - 15 x - 5 \cdot 2$

Step 7.1.5

Multiply $- 5$ by $2$ .

$3 x^{2} + 2 x - 15 x - 10$

Step 7.2

Subtract $15 x$ from $2 x$ .

$3 x^{2} - 13 x - 10$