Algebra Examples

$\frac{x^{2} + 8 x + 15}{- 5 x - 5} \cdot \frac{x^{2} + 2 x + 1}{2 x + 10}$

Step 1

Factor $x^{2} + 8 x + 15$ using the AC method.

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Step 1.1

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 $8$ .

$3, 5$

Step 1.2

Write the factored form using these integers.

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

Step 2

Factor $5$ out of $- 5 x - 5$ .

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Step 2.1

Factor $5$ out of $- 5 x$ .

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

Step 2.2

Factor $5$ out of $- 5$ .

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

Step 2.3

Factor $5$ out of $5 (- x) + 5 (- 1)$ .

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

Step 3

Factor using the perfect square rule.

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Step 3.1

Rewrite $1$ as $1^{2}$ .

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

Step 3.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$2 x = 2 \cdot x \cdot 1$

Step 3.3

Rewrite the polynomial.

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

Step 3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = x$ and $b = 1$ .

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

Step 4

Simplify terms.

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Step 4.1

Factor $2$ out of $2 x + 10$ .

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Step 4.1.1

Factor $2$ out of $2 x$ .

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

Step 4.1.2

Factor $2$ out of $10$ .

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

Step 4.1.3

Factor $2$ out of $2 x + 2 \cdot 5$ .

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

Step 4.2

Combine.

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

Step 4.3

Cancel the common factor of $x + 5$ .

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 4.4

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

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Step 4.4.1

Factor $- 1$ out of $x$ .

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

Step 4.4.2

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

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

Step 4.4.3

Factor $- 1$ out of $- 1 (- x) - 1 (- 1)$ .

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

Step 4.4.4

Apply the product rule to $- 1 (- x - 1)$ .

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

Step 4.4.5

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

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

Step 4.4.6

Multiply ${(- x - 1)}^{2}$ by $1$ .

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

Step 4.4.7

Factor $- x - 1$ out of $(x + 3) {(- x - 1)}^{2}$ .

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

Step 4.4.8

Cancel the common factors.

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Step 4.4.8.1

Factor $- x - 1$ out of $5 (- x - 1) \cdot 2$ .

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

Step 4.4.8.2

Cancel the common factor.

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

Step 4.4.8.3

Rewrite the expression.

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

Step 4.5

Multiply $5$ by $2$ .

$\frac{(x + 3) (- x - 1)}{10}$

Step 4.6

Factor $- 1$ out of $- x$ .

$\frac{(x + 3) (- (x) - 1)}{10}$

Step 4.7

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

$\frac{(x + 3) (- (x) - 1 (1))}{10}$

Step 4.8

Factor $- 1$ out of $- (x) - 1 (1)$ .

$\frac{(x + 3) (- (x + 1))}{10}$

Step 4.9

Simplify the expression.

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Step 4.9.1

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

$\frac{(x + 3) (- 1 (x + 1))}{10}$

Step 4.9.2

Move the negative in front of the fraction.

$- \frac{(x + 3) (x + 1)}{10}$