Precalculus Examples

$- \frac{7}{32} \cdot ((x + 8) {(x + 1)}^{2} (x - 4))$

Step 1

Write $- \frac{7}{32} \cdot ((x + 8) {(x + 1)}^{2} (x - 4))$ as an equation.

$y = - \frac{7}{32} \cdot ((x + 8) {(x + 1)}^{2} (x - 4))$

Step 2

Find the x-intercepts.

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Step 2.1

To find the x-intercept(s), substitute in $0$ for $y$ and solve for $x$ .

$0 = - \frac{7}{32} \cdot ((x + 8) {(x + 1)}^{2} (x - 4))$

Step 2.2

Solve the equation.

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Step 2.2.1

Rewrite the equation as $- \frac{7}{32} \cdot ((x + 8) {(x + 1)}^{2} (x - 4)) = 0$ .

$- \frac{7}{32} \cdot ((x + 8) {(x + 1)}^{2} (x - 4)) = 0$

Step 2.2.2

Multiply both sides of the equation by $- \frac{32}{7}$ .

$- \frac{32}{7} (- \frac{7}{32} \cdot ((x + 8) {(x + 1)}^{2} (x - 4))) = - \frac{32}{7} \cdot 0$

Step 2.2.3

Simplify both sides of the equation.

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Step 2.2.3.1

Simplify the left side.

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Step 2.2.3.1.1

Simplify $- \frac{32}{7} (- \frac{7}{32} \cdot ((x + 8) {(x + 1)}^{2} (x - 4)))$ .

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Step 2.2.3.1.1.1

Apply the distributive property.

$- \frac{32}{7} (- \frac{7}{32} \cdot ((x {(x + 1)}^{2} + 8 {(x + 1)}^{2}) (x - 4))) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.2

Expand $(x {(x + 1)}^{2} + 8 {(x + 1)}^{2}) (x - 4)$ using the FOIL Method.

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Step 2.2.3.1.1.2.1

Apply the distributive property.

$- \frac{32}{7} (- \frac{7}{32} \cdot (x {(x + 1)}^{2} (x - 4) + 8 {(x + 1)}^{2} (x - 4))) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.2.2

Apply the distributive property.

$- \frac{32}{7} (- \frac{7}{32} \cdot (x {(x + 1)}^{2} x + x {(x + 1)}^{2} \cdot - 4 + 8 {(x + 1)}^{2} (x - 4))) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.2.3

Apply the distributive property.

$- \frac{32}{7} (- \frac{7}{32} \cdot (x {(x + 1)}^{2} x + x {(x + 1)}^{2} \cdot - 4 + 8 {(x + 1)}^{2} x + 8 {(x + 1)}^{2} \cdot - 4)) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.3

Simplify and combine like terms.

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Step 2.2.3.1.1.3.1

Simplify each term.

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Step 2.2.3.1.1.3.1.1

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

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Step 2.2.3.1.1.3.1.1.1

Move $x$ .

$- \frac{32}{7} (- \frac{7}{32} \cdot (x \cdot x {(x + 1)}^{2} + x {(x + 1)}^{2} \cdot - 4 + 8 {(x + 1)}^{2} x + 8 {(x + 1)}^{2} \cdot - 4)) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.3.1.1.2

Multiply $x$ by $x$ .

$- \frac{32}{7} (- \frac{7}{32} \cdot (x^{2} {(x + 1)}^{2} + x {(x + 1)}^{2} \cdot - 4 + 8 {(x + 1)}^{2} x + 8 {(x + 1)}^{2} \cdot - 4)) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.3.1.2

Move $- 4$ to the left of $x {(x + 1)}^{2}$ .

$- \frac{32}{7} (- \frac{7}{32} \cdot (x^{2} {(x + 1)}^{2} - 4 \cdot (x {(x + 1)}^{2}) + 8 {(x + 1)}^{2} x + 8 {(x + 1)}^{2} \cdot - 4)) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.3.1.3

Multiply $- 4$ by $8$ .

$- \frac{32}{7} (- \frac{7}{32} \cdot (x^{2} {(x + 1)}^{2} - 4 x {(x + 1)}^{2} + 8 {(x + 1)}^{2} x - 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.3.2

Add $- 4 x {(x + 1)}^{2}$ and $8 {(x + 1)}^{2} x$ .

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Step 2.2.3.1.1.3.2.1

Move ${(x + 1)}^{2}$ .

$- \frac{32}{7} (- \frac{7}{32} \cdot (x^{2} {(x + 1)}^{2} - 4 x {(x + 1)}^{2} + 8 x {(x + 1)}^{2} - 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.3.2.2

Add $- 4 x {(x + 1)}^{2}$ and $8 x {(x + 1)}^{2}$ .

$- \frac{32}{7} (- \frac{7}{32} \cdot (x^{2} {(x + 1)}^{2} + 4 x {(x + 1)}^{2} - 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.4

Apply the distributive property.

$- \frac{32}{7} (- \frac{7}{32} (x^{2} {(x + 1)}^{2}) - \frac{7}{32} (4 x {(x + 1)}^{2}) - \frac{7}{32} (- 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5

Simplify.

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Step 2.2.3.1.1.5.1

Multiply $- \frac{7}{32} (x^{2} {(x + 1)}^{2})$ .

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Step 2.2.3.1.1.5.1.1

Combine $x^{2}$ and $\frac{7}{32}$ .

$- \frac{32}{7} (- \frac{x^{2} \cdot 7}{32} {(x + 1)}^{2} - \frac{7}{32} (4 x {(x + 1)}^{2}) - \frac{7}{32} (- 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.1.2

Combine ${(x + 1)}^{2}$ and $\frac{x^{2} \cdot 7}{32}$ .

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} - \frac{7}{32} (4 x {(x + 1)}^{2}) - \frac{7}{32} (- 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.2

Cancel the common factor of $4$ .

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Step 2.2.3.1.1.5.2.1

Move the leading negative in $- \frac{7}{32}$ into the numerator.

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{- 7}{32} (4 x {(x + 1)}^{2}) - \frac{7}{32} (- 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.2.2

Factor $4$ out of $32$ .

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{- 7}{4 (8)} (4 x {(x + 1)}^{2}) - \frac{7}{32} (- 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.2.3

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

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{- 7}{4 (8)} (4 (x {(x + 1)}^{2})) - \frac{7}{32} (- 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.2.4

Cancel the common factor.

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{- 7}{4 \cdot 8} (4 (x {(x + 1)}^{2})) - \frac{7}{32} (- 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.2.5

Rewrite the expression.

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{- 7}{8} (x {(x + 1)}^{2}) - \frac{7}{32} (- 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.3

Combine $x$ and $\frac{- 7}{8}$ .

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{x \cdot - 7}{8} {(x + 1)}^{2} - \frac{7}{32} (- 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.4

Combine $\frac{x \cdot - 7}{8}$ and ${(x + 1)}^{2}$ .

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{x \cdot - 7 {(x + 1)}^{2}}{8} - \frac{7}{32} (- 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.5

Cancel the common factor of $32$ .

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Step 2.2.3.1.1.5.5.1

Move the leading negative in $- \frac{7}{32}$ into the numerator.

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{x \cdot - 7 {(x + 1)}^{2}}{8} + \frac{- 7}{32} (- 32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.5.2

Factor $32$ out of $- 32 {(x + 1)}^{2}$ .

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{x \cdot - 7 {(x + 1)}^{2}}{8} + \frac{- 7}{32} (32 (- {(x + 1)}^{2}))) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.5.3

Cancel the common factor.

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{x \cdot - 7 {(x + 1)}^{2}}{8} + \frac{- 7}{32} (32 (- {(x + 1)}^{2}))) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.5.4

Rewrite the expression.

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{x \cdot - 7 {(x + 1)}^{2}}{8} - 7 (- {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.5.6

Multiply $- 1$ by $- 7$ .

$- \frac{32}{7} (- \frac{{(x + 1)}^{2} (x^{2} \cdot 7)}{32} + \frac{x \cdot - 7 {(x + 1)}^{2}}{8} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.6

Simplify each term.

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Step 2.2.3.1.1.6.1

Move $7$ to the left of ${(x + 1)}^{2} x^{2}$ .

$- \frac{32}{7} (- \frac{7 \cdot ({(x + 1)}^{2} x^{2})}{32} + \frac{x \cdot - 7 {(x + 1)}^{2}}{8} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.6.2

Move $- 7$ to the left of $x$ .

$- \frac{32}{7} (- \frac{7 {(x + 1)}^{2} x^{2}}{32} + \frac{- 7 \cdot x {(x + 1)}^{2}}{8} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.6.3

Move the negative in front of the fraction.

$- \frac{32}{7} (- \frac{7 {(x + 1)}^{2} x^{2}}{32} - \frac{7 x {(x + 1)}^{2}}{8} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.7

To write $- \frac{7 x {(x + 1)}^{2}}{8}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$- \frac{32}{7} (- \frac{7 {(x + 1)}^{2} x^{2}}{32} - \frac{7 x {(x + 1)}^{2}}{8} \cdot \frac{4}{4} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.8

Write each expression with a common denominator of $32$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.3.1.1.8.1

Multiply $\frac{7 x {(x + 1)}^{2}}{8}$ by $\frac{4}{4}$ .

$- \frac{32}{7} (- \frac{7 {(x + 1)}^{2} x^{2}}{32} - \frac{7 x {(x + 1)}^{2} \cdot 4}{8 \cdot 4} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.8.2

Multiply $8$ by $4$ .

$- \frac{32}{7} (- \frac{7 {(x + 1)}^{2} x^{2}}{32} - \frac{7 x {(x + 1)}^{2} \cdot 4}{32} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.9

Combine the numerators over the common denominator.

$- \frac{32}{7} (\frac{- 7 {(x + 1)}^{2} x^{2} - 7 x {(x + 1)}^{2} \cdot 4}{32} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.10

Simplify the numerator.

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Step 2.2.3.1.1.10.1

Factor $7 {(x + 1)}^{2} x$ out of $- 7 {(x + 1)}^{2} x^{2} - 7 x {(x + 1)}^{2} \cdot 4$ .

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Step 2.2.3.1.1.10.1.1

Factor $7 {(x + 1)}^{2} x$ out of $- 7 {(x + 1)}^{2} x^{2}$ .

$- \frac{32}{7} (\frac{7 {(x + 1)}^{2} x (- x) - 7 x {(x + 1)}^{2} \cdot 4}{32} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.10.1.2

Factor $7 {(x + 1)}^{2} x$ out of $- 7 x {(x + 1)}^{2} \cdot 4$ .

$- \frac{32}{7} (\frac{7 {(x + 1)}^{2} x (- x) + 7 {(x + 1)}^{2} x (- 1 \cdot 4)}{32} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.10.1.3

Factor $7 {(x + 1)}^{2} x$ out of $7 {(x + 1)}^{2} x (- x) + 7 {(x + 1)}^{2} x (- 1 \cdot 4)$ .

$- \frac{32}{7} (\frac{7 {(x + 1)}^{2} x (- x - 1 \cdot 4)}{32} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.10.2

Multiply $- 1$ by $4$ .

$- \frac{32}{7} (\frac{7 {(x + 1)}^{2} x (- x - 4)}{32} + 7 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.11

To write $7 {(x + 1)}^{2}$ as a fraction with a common denominator, multiply by $\frac{32}{32}$ .

$- \frac{32}{7} (\frac{7 {(x + 1)}^{2} x (- x - 4)}{32} + 7 {(x + 1)}^{2} \cdot \frac{32}{32}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.12

Simplify terms.

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Step 2.2.3.1.1.12.1

Combine $7 {(x + 1)}^{2}$ and $\frac{32}{32}$ .

$- \frac{32}{7} (\frac{7 {(x + 1)}^{2} x (- x - 4)}{32} + \frac{7 {(x + 1)}^{2} \cdot 32}{32}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.12.2

Combine the numerators over the common denominator.

$- \frac{32}{7} \cdot \frac{7 {(x + 1)}^{2} x (- x - 4) + 7 {(x + 1)}^{2} \cdot 32}{32} = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.12.3

Cancel the common factor of $32$ .

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Step 2.2.3.1.1.12.3.1

Move the leading negative in $- \frac{32}{7}$ into the numerator.

$\frac{- 32}{7} \cdot \frac{7 {(x + 1)}^{2} x (- x - 4) + 7 {(x + 1)}^{2} \cdot 32}{32} = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.12.3.2

Factor $32$ out of $- 32$ .

$\frac{32 (- 1)}{7} \cdot \frac{7 {(x + 1)}^{2} x (- x - 4) + 7 {(x + 1)}^{2} \cdot 32}{32} = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.12.3.3

Cancel the common factor.

$\frac{32 \cdot - 1}{7} \cdot \frac{7 {(x + 1)}^{2} x (- x - 4) + 7 {(x + 1)}^{2} \cdot 32}{32} = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.12.3.4

Rewrite the expression.

$\frac{- 1}{7} (7 {(x + 1)}^{2} x (- x - 4) + 7 {(x + 1)}^{2} \cdot 32) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.12.4

Simplify the expression.

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Step 2.2.3.1.1.12.4.1

Multiply $32$ by $7$ .

$\frac{- 1}{7} (7 {(x + 1)}^{2} x (- x - 4) + 224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.12.4.2

Move the negative in front of the fraction.

$- \frac{1}{7} (7 {(x + 1)}^{2} x (- x - 4) + 224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.13

Simplify each term.

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Step 2.2.3.1.1.13.1

Apply the distributive property.

$- \frac{1}{7} (7 {(x + 1)}^{2} x (- x) + 7 {(x + 1)}^{2} x \cdot - 4 + 224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.13.2

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

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Step 2.2.3.1.1.13.2.1

Move $x$ .

$- \frac{1}{7} (7 {(x + 1)}^{2} (x \cdot x) \cdot - 1 + 7 {(x + 1)}^{2} x \cdot - 4 + 224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.13.2.2

Multiply $x$ by $x$ .

$- \frac{1}{7} (7 {(x + 1)}^{2} x^{2} \cdot - 1 + 7 {(x + 1)}^{2} x \cdot - 4 + 224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.13.3

Multiply $- 4$ by $7$ .

$- \frac{1}{7} (7 {(x + 1)}^{2} x^{2} \cdot - 1 - 28 {(x + 1)}^{2} x + 224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.13.4

Multiply $- 1$ by $7$ .

$- \frac{1}{7} (- 7 {(x + 1)}^{2} x^{2} - 28 {(x + 1)}^{2} x + 224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.14

Apply the distributive property.

$- \frac{1}{7} (- 7 {(x + 1)}^{2} x^{2}) - \frac{1}{7} (- 28 {(x + 1)}^{2} x) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15

Simplify.

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Step 2.2.3.1.1.15.1

Cancel the common factor of $7$ .

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Step 2.2.3.1.1.15.1.1

Move the leading negative in $- \frac{1}{7}$ into the numerator.

$\frac{- 1}{7} (- 7 {(x + 1)}^{2} x^{2}) - \frac{1}{7} (- 28 {(x + 1)}^{2} x) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.1.2

Factor $7$ out of $- 7 {(x + 1)}^{2} x^{2}$ .

$\frac{- 1}{7} (7 (- {(x + 1)}^{2} x^{2})) - \frac{1}{7} (- 28 {(x + 1)}^{2} x) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.1.3

Cancel the common factor.

$\frac{- 1}{7} (7 (- {(x + 1)}^{2} x^{2})) - \frac{1}{7} (- 28 {(x + 1)}^{2} x) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.1.4

Rewrite the expression.

$- (- {(x + 1)}^{2} x^{2}) - \frac{1}{7} (- 28 {(x + 1)}^{2} x) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.2

Multiply $- 1$ by $- 1$ .

$1 ({(x + 1)}^{2} x^{2}) - \frac{1}{7} (- 28 {(x + 1)}^{2} x) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.3

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

${(x + 1)}^{2} x^{2} - \frac{1}{7} (- 28 {(x + 1)}^{2} x) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.4

Cancel the common factor of $7$ .

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Step 2.2.3.1.1.15.4.1

Move the leading negative in $- \frac{1}{7}$ into the numerator.

${(x + 1)}^{2} x^{2} + \frac{- 1}{7} (- 28 {(x + 1)}^{2} x) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.4.2

Factor $7$ out of $- 28 {(x + 1)}^{2} x$ .

${(x + 1)}^{2} x^{2} + \frac{- 1}{7} (7 (- 4 {(x + 1)}^{2} x)) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.4.3

Cancel the common factor.

${(x + 1)}^{2} x^{2} + \frac{- 1}{7} (7 (- 4 {(x + 1)}^{2} x)) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.4.4

Rewrite the expression.

${(x + 1)}^{2} x^{2} - (- 4 {(x + 1)}^{2} x) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.5

Multiply $- 4$ by $- 1$ .

${(x + 1)}^{2} x^{2} + 4 ({(x + 1)}^{2} x) - \frac{1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.6

Cancel the common factor of $7$ .

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Step 2.2.3.1.1.15.6.1

Move the leading negative in $- \frac{1}{7}$ into the numerator.

${(x + 1)}^{2} x^{2} + 4 ({(x + 1)}^{2} x) + \frac{- 1}{7} (224 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.6.2

Factor $7$ out of $224 {(x + 1)}^{2}$ .

${(x + 1)}^{2} x^{2} + 4 ({(x + 1)}^{2} x) + \frac{- 1}{7} (7 (32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.6.3

Cancel the common factor.

${(x + 1)}^{2} x^{2} + 4 ({(x + 1)}^{2} x) + \frac{- 1}{7} (7 (32 {(x + 1)}^{2})) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.6.4

Rewrite the expression.

${(x + 1)}^{2} x^{2} + 4 ({(x + 1)}^{2} x) - (32 {(x + 1)}^{2}) = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.15.7

Multiply $32$ by $- 1$ .

${(x + 1)}^{2} x^{2} + 4 ({(x + 1)}^{2} x) - 32 {(x + 1)}^{2} = - \frac{32}{7} \cdot 0$

Step 2.2.3.1.1.16

Reorder factors in ${(x + 1)}^{2} x^{2} + 4 {(x + 1)}^{2} x - 32 {(x + 1)}^{2}$ .

$x^{2} {(x + 1)}^{2} + 4 x {(x + 1)}^{2} - 32 {(x + 1)}^{2} = - \frac{32}{7} \cdot 0$

Step 2.2.3.2

Simplify the right side.

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Step 2.2.3.2.1

Multiply $- \frac{32}{7} \cdot 0$ .

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Step 2.2.3.2.1.1

Multiply $0$ by $- 1$ .

$x^{2} {(x + 1)}^{2} + 4 x {(x + 1)}^{2} - 32 {(x + 1)}^{2} = 0 (\frac{32}{7})$

Step 2.2.3.2.1.2

Multiply $0$ by $\frac{32}{7}$ .

$x^{2} {(x + 1)}^{2} + 4 x {(x + 1)}^{2} - 32 {(x + 1)}^{2} = 0$

Step 2.2.4

Factor the left side of the equation.

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Step 2.2.4.1

Factor ${(x + 1)}^{2}$ out of $x^{2} {(x + 1)}^{2} + 4 x {(x + 1)}^{2} - 32 {(x + 1)}^{2}$ .

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Step 2.2.4.1.1

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

${(x + 1)}^{2} x^{2} + 4 x {(x + 1)}^{2} - 32 {(x + 1)}^{2} = 0$

Step 2.2.4.1.2

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

${(x + 1)}^{2} x^{2} + {(x + 1)}^{2} (4 x) - 32 {(x + 1)}^{2} = 0$

Step 2.2.4.1.3

Factor ${(x + 1)}^{2}$ out of $- 32 {(x + 1)}^{2}$ .

${(x + 1)}^{2} x^{2} + {(x + 1)}^{2} (4 x) + {(x + 1)}^{2} \cdot - 32 = 0$

Step 2.2.4.1.4

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

${(x + 1)}^{2} (x^{2} + 4 x) + {(x + 1)}^{2} \cdot - 32 = 0$

Step 2.2.4.1.5

Factor ${(x + 1)}^{2}$ out of ${(x + 1)}^{2} (x^{2} + 4 x) + {(x + 1)}^{2} \cdot - 32$ .

${(x + 1)}^{2} (x^{2} + 4 x - 32) = 0$

Step 2.2.4.2

Factor.

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Step 2.2.4.2.1

Factor $x^{2} + 4 x - 32$ using the AC method.

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Step 2.2.4.2.1.1

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 $- 32$ and whose sum is $4$ .

$- 4, 8$

Step 2.2.4.2.1.2

Write the factored form using these integers.

${(x + 1)}^{2} ((x - 4) (x + 8)) = 0$

Step 2.2.4.2.2

Remove unnecessary parentheses.

${(x + 1)}^{2} (x - 4) (x + 8) = 0$

Step 2.2.5

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

${(x + 1)}^{2} = 0$

$x - 4 = 0$

$x + 8 = 0$

Step 2.2.6

Set ${(x + 1)}^{2}$ equal to $0$ and solve for $x$ .

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Step 2.2.6.1

Set ${(x + 1)}^{2}$ equal to $0$ .

${(x + 1)}^{2} = 0$

Step 2.2.6.2

Solve ${(x + 1)}^{2} = 0$ for $x$ .

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Step 2.2.6.2.1

Set the $x + 1$ equal to $0$ .

$x + 1 = 0$

Step 2.2.6.2.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.2.7

Set $x - 4$ equal to $0$ and solve for $x$ .

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Step 2.2.7.1

Set $x - 4$ equal to $0$ .

$x - 4 = 0$

Step 2.2.7.2

Add $4$ to both sides of the equation.

$x = 4$

Step 2.2.8

Set $x + 8$ equal to $0$ and solve for $x$ .

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Step 2.2.8.1

Set $x + 8$ equal to $0$ .

$x + 8 = 0$

Step 2.2.8.2

Subtract $8$ from both sides of the equation.

$x = - 8$

Step 2.2.9

The final solution is all the values that make ${(x + 1)}^{2} (x - 4) (x + 8) = 0$ true.

$x = - 1, 4, - 8$

Step 2.3

x-intercept(s) in point form.

x-intercept(s): $(- 1, 0), (4, 0), (- 8, 0)$

Step 3

Find the y-intercepts.

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Step 3.1

To find the y-intercept(s), substitute in $0$ for $x$ and solve for $y$ .

$y = - \frac{7}{32} \cdot (((0) + 8) {((0) + 1)}^{2} ((0) - 4))$

Step 3.2

Solve the equation.

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Step 3.2.1

Remove parentheses.

$y = - \frac{7}{32} \cdot ((0 + 8) {((0) + 1)}^{2} ((0) - 4))$

Step 3.2.2

Remove parentheses.

$y = - \frac{7}{32} \cdot ((0 + 8) {(0 + 1)}^{2} ((0) - 4))$

Step 3.2.3

Remove parentheses.

$y = - \frac{7}{32} \cdot ((0 + 8) {(0 + 1)}^{2} (0 - 4))$

Step 3.2.4

Remove parentheses.

$y = - \frac{7}{32} \cdot (((0 + 8) {(0 + 1)}^{2}) (0 - 4))$

Step 3.2.5

Remove parentheses.

$y = - \frac{7}{32} \cdot ((0 + 8) {(0 + 1)}^{2} (0 - 4))$

Step 3.2.6

Remove parentheses.

$y = - \frac{7}{32} \cdot (((0) + 8) {((0) + 1)}^{2} ((0) - 4))$

Step 3.2.7

Simplify $- \frac{7}{32} \cdot (((0) + 8) {((0) + 1)}^{2} ((0) - 4))$ .

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Step 3.2.7.1

Simplify the expression.

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Step 3.2.7.1.1

Add $0$ and $8$ .

$y = - \frac{7}{32} \cdot (8 {((0) + 1)}^{2} ((0) - 4))$

Step 3.2.7.1.2

Add $0$ and $1$ .

$y = - \frac{7}{32} \cdot (8 \cdot 1^{2} ((0) - 4))$

Step 3.2.7.1.3

One to any power is one.

$y = - \frac{7}{32} \cdot (8 \cdot 1 ((0) - 4))$

Step 3.2.7.1.4

Multiply $8$ by $1$ .

$y = - \frac{7}{32} \cdot (8 ((0) - 4))$

Step 3.2.7.1.5

Subtract $4$ from $0$ .

$y = - \frac{7}{32} \cdot (8 \cdot - 4)$

Step 3.2.7.1.6

Multiply $8$ by $- 4$ .

$y = - \frac{7}{32} \cdot - 32$

Step 3.2.7.2

Cancel the common factor of $32$ .

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Step 3.2.7.2.1

Move the leading negative in $- \frac{7}{32}$ into the numerator.

$y = \frac{- 7}{32} \cdot - 32$

Step 3.2.7.2.2

Factor $32$ out of $- 32$ .

$y = \frac{- 7}{32} \cdot (32 (- 1))$

Step 3.2.7.2.3

Cancel the common factor.

$y = \frac{- 7}{32} \cdot (32 \cdot - 1)$

Step 3.2.7.2.4

Rewrite the expression.

$y = - 7 \cdot - 1$

Step 3.2.7.3

Multiply $- 7$ by $- 1$ .

$y = 7$

Step 3.3

y-intercept(s) in point form.

y-intercept(s): $(0, 7)$

Step 4

List the intersections.

x-intercept(s): $(- 1, 0), (4, 0), (- 8, 0)$

y-intercept(s): $(0, 7)$

Step 5