Algebra Examples

$\frac{4 x^{2} + 23 x + 15}{16 x^{2} - 9} \cdot \frac{4 x^{2} + 9 x - 9}{x^{2} + 2 x - 15}$

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 $a \cdot c = 4 \cdot 15 = 60$ and whose sum is $b = 23$ .

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Step 1.1.1

Factor $23$ out of $23 x$ .

$\frac{4 x^{2} + 23 (x) + 15}{16 x^{2} - 9} \cdot \frac{4 x^{2} + 9 x - 9}{x^{2} + 2 x - 15}$

Step 1.1.2

Rewrite $23$ as $3$ plus $20$

$\frac{4 x^{2} + (3 + 20) x + 15}{16 x^{2} - 9} \cdot \frac{4 x^{2} + 9 x - 9}{x^{2} + 2 x - 15}$

Step 1.1.3

Apply the distributive property.

$\frac{4 x^{2} + 3 x + 20 x + 15}{16 x^{2} - 9} \cdot \frac{4 x^{2} + 9 x - 9}{x^{2} + 2 x - 15}$

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.

$\frac{(4 x^{2} + 3 x) + 20 x + 15}{16 x^{2} - 9} \cdot \frac{4 x^{2} + 9 x - 9}{x^{2} + 2 x - 15}$

Step 1.2.2

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

$\frac{x (4 x + 3) + 5 (4 x + 3)}{16 x^{2} - 9} \cdot \frac{4 x^{2} + 9 x - 9}{x^{2} + 2 x - 15}$

Step 1.3

Factor the polynomial by factoring out the greatest common factor, $4 x + 3$ .

$\frac{(4 x + 3) (x + 5)}{16 x^{2} - 9} \cdot \frac{4 x^{2} + 9 x - 9}{x^{2} + 2 x - 15}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $16 x^{2}$ as ${(4 x)}^{2}$ .

$\frac{(4 x + 3) (x + 5)}{{(4 x)}^{2} - 9} \cdot \frac{4 x^{2} + 9 x - 9}{x^{2} + 2 x - 15}$

Step 2.2

Rewrite $9$ as $3^{2}$ .

$\frac{(4 x + 3) (x + 5)}{{(4 x)}^{2} - 3^{2}} \cdot \frac{4 x^{2} + 9 x - 9}{x^{2} + 2 x - 15}$

Step 2.3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 4 x$ and $b = 3$ .

$\frac{(4 x + 3) (x + 5)}{(4 x + 3) (4 x - 3)} \cdot \frac{4 x^{2} + 9 x - 9}{x^{2} + 2 x - 15}$

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 $a \cdot c = 4 \cdot - 9 = - 36$ and whose sum is $b = 9$ .

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Step 3.1.1

Factor $9$ out of $9 x$ .

$\frac{(4 x + 3) (x + 5)}{(4 x + 3) (4 x - 3)} \cdot \frac{4 x^{2} + 9 (x) - 9}{x^{2} + 2 x - 15}$

Step 3.1.2

Rewrite $9$ as $- 3$ plus $12$

$\frac{(4 x + 3) (x + 5)}{(4 x + 3) (4 x - 3)} \cdot \frac{4 x^{2} + (- 3 + 12) x - 9}{x^{2} + 2 x - 15}$

Step 3.1.3

Apply the distributive property.

$\frac{(4 x + 3) (x + 5)}{(4 x + 3) (4 x - 3)} \cdot \frac{4 x^{2} - 3 x + 12 x - 9}{x^{2} + 2 x - 15}$

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.

$\frac{(4 x + 3) (x + 5)}{(4 x + 3) (4 x - 3)} \cdot \frac{(4 x^{2} - 3 x) + 12 x - 9}{x^{2} + 2 x - 15}$

Step 3.2.2

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

$\frac{(4 x + 3) (x + 5)}{(4 x + 3) (4 x - 3)} \cdot \frac{x (4 x - 3) + 3 (4 x - 3)}{x^{2} + 2 x - 15}$

Step 3.3

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

$\frac{(4 x + 3) (x + 5)}{(4 x + 3) (4 x - 3)} \cdot \frac{(4 x - 3) (x + 3)}{x^{2} + 2 x - 15}$

Step 4

Factor $x^{2} + 2 x - 15$ using the AC method.

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Step 4.1

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 $- 15$ and whose sum is $2$ .

$- 3, 5$

Step 4.2

Write the factored form using these integers.

$\frac{(4 x + 3) (x + 5)}{(4 x + 3) (4 x - 3)} \cdot \frac{(4 x - 3) (x + 3)}{(x - 3) (x + 5)}$

Step 5

Simplify terms.

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Step 5.1

Cancel the common factor of $x + 5$ .

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Step 5.1.1

Factor $x + 5$ out of $(4 x + 3) (x + 5)$ .

$\frac{(x + 5) (4 x + 3)}{(4 x + 3) (4 x - 3)} \cdot \frac{(4 x - 3) (x + 3)}{(x - 3) (x + 5)}$

Step 5.1.2

Factor $x + 5$ out of $(x - 3) (x + 5)$ .

$\frac{(x + 5) (4 x + 3)}{(4 x + 3) (4 x - 3)} \cdot \frac{(4 x - 3) (x + 3)}{(x + 5) (x - 3)}$

Step 5.1.3

Cancel the common factor.

$\frac{(x + 5) (4 x + 3)}{(4 x + 3) (4 x - 3)} \cdot \frac{(4 x - 3) (x + 3)}{(x + 5) (x - 3)}$

Step 5.1.4

Rewrite the expression.

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

Step 5.2

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

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Step 5.2.1

Factor $4 x - 3$ out of $(4 x + 3) (4 x - 3)$ .

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

Step 5.2.2

Cancel the common factor.

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

Step 5.2.3

Rewrite the expression.

$\frac{4 x + 3}{4 x + 3} \cdot \frac{x + 3}{x - 3}$

Step 5.3

Multiply $\frac{4 x + 3}{4 x + 3}$ by $\frac{x + 3}{x - 3}$ .

$\frac{(4 x + 3) (x + 3)}{(4 x + 3) (x - 3)}$

Step 5.4

Cancel the common factor of $4 x + 3$ .

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Step 5.4.1

Cancel the common factor.

$\frac{(4 x + 3) (x + 3)}{(4 x + 3) (x - 3)}$

Step 5.4.2

Rewrite the expression.

$\frac{x + 3}{x - 3}$