Algebra Examples

$f (x) = - (x - 1) (x + 2) {(x + 1)}^{2}$

Step 1

Identify the degree of the function.

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Step 1.1

Simplify and reorder the polynomial.

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Step 1.1.1

Simplify by multiplying through.

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Step 1.1.1.1

Apply the distributive property.

$(- x - - 1) (x + 2) {(x + 1)}^{2}$

Step 1.1.1.2

Multiply $- 1$ by $- 1$ .

$(- x + 1) (x + 2) {(x + 1)}^{2}$

Step 1.1.2

Expand $(- x + 1) (x + 2)$ using the FOIL Method.

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Step 1.1.2.1

Apply the distributive property.

$(- x (x + 2) + 1 (x + 2)) {(x + 1)}^{2}$

Step 1.1.2.2

Apply the distributive property.

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

Step 1.1.2.3

Apply the distributive property.

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

Step 1.1.3

Simplify and combine like terms.

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Step 1.1.3.1

Simplify each term.

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Step 1.1.3.1.1

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

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Step 1.1.3.1.1.1

Move $x$ .

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

Step 1.1.3.1.1.2

Multiply $x$ by $x$ .

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

Step 1.1.3.1.2

Multiply $2$ by $- 1$ .

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

Step 1.1.3.1.3

Multiply $x$ by $1$ .

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

Step 1.1.3.1.4

Multiply $2$ by $1$ .

$(- x^{2} - 2 x + x + 2) {(x + 1)}^{2}$

Step 1.1.3.2

Add $- 2 x$ and $x$ .

$(- x^{2} - x + 2) {(x + 1)}^{2}$

Step 1.1.4

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$(- x^{2} - x + 2) ((x + 1) (x + 1))$

Step 1.1.5

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 1.1.5.1

Apply the distributive property.

$(- x^{2} - x + 2) (x (x + 1) + 1 (x + 1))$

Step 1.1.5.2

Apply the distributive property.

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

Step 1.1.5.3

Apply the distributive property.

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

Step 1.1.6

Simplify and combine like terms.

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Step 1.1.6.1

Simplify each term.

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Step 1.1.6.1.1

Multiply $x$ by $x$ .

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

Step 1.1.6.1.2

Multiply $x$ by $1$ .

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

Step 1.1.6.1.3

Multiply $x$ by $1$ .

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

Step 1.1.6.1.4

Multiply $1$ by $1$ .

$(- x^{2} - x + 2) (x^{2} + x + x + 1)$

Step 1.1.6.2

Add $x$ and $x$ .

$(- x^{2} - x + 2) (x^{2} + 2 x + 1)$

Step 1.1.7

Expand $(- x^{2} - x + 2) (x^{2} + 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 1.1.8

Simplify terms.

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Step 1.1.8.1

Simplify each term.

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Step 1.1.8.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 1.1.8.1.1.1

Move $x^{2}$ .

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

Step 1.1.8.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.8.1.1.3

Add $2$ and $2$ .

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

Step 1.1.8.1.2

Rewrite using the commutative property of multiplication.

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

Step 1.1.8.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 1.1.8.1.3.1

Move $x$ .

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

Step 1.1.8.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 1.1.8.1.3.2.1

Raise $x$ to the power of $1$ .

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

Step 1.1.8.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 1.1.8.1.3.3

Add $1$ and $2$ .

$- x^{4} - 1 \cdot 2 x^{3} - x^{2} \cdot 1 - x \cdot x^{2} - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.4

Multiply $- 1$ by $2$ .

$- x^{4} - 2 x^{3} - x^{2} \cdot 1 - x \cdot x^{2} - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.5

Multiply $- 1$ by $1$ .

$- x^{4} - 2 x^{3} - x^{2} - x \cdot x^{2} - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.6

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.1.8.1.6.1

Move $x^{2}$ .

$- x^{4} - 2 x^{3} - x^{2} - (x^{2} x) - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.6.2

Multiply $x^{2}$ by $x$ .

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Step 1.1.8.1.6.2.1

Raise $x$ to the power of $1$ .

$- x^{4} - 2 x^{3} - x^{2} - (x^{2} x^{1}) - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- x^{4} - 2 x^{3} - x^{2} - x^{2 + 1} - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.6.3

Add $2$ and $1$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.7

Rewrite using the commutative property of multiplication.

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 1 \cdot 2 x \cdot x - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.8

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

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Step 1.1.8.1.8.1

Move $x$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 1 \cdot 2 (x \cdot x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.8.2

Multiply $x$ by $x$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 1 \cdot 2 x^{2} - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.9

Multiply $- 1$ by $2$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 2 x^{2} - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.10

Multiply $- 1$ by $1$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 2 x^{2} - x + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 1.1.8.1.11

Multiply $2$ by $2$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 2 x^{2} - x + 2 x^{2} + 4 x + 2 \cdot 1$

Step 1.1.8.1.12

Multiply $2$ by $1$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 2 x^{2} - x + 2 x^{2} + 4 x + 2$

Step 1.1.8.2

Simplify by adding terms.

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Step 1.1.8.2.1

Combine the opposite terms in $- x^{4} - 2 x^{3} - x^{2} - x^{3} - 2 x^{2} - x + 2 x^{2} + 4 x + 2$ .

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Step 1.1.8.2.1.1

Add $- 2 x^{2}$ and $2 x^{2}$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - x + 0 + 4 x + 2$

Step 1.1.8.2.1.2

Add $- x^{4} - 2 x^{3} - x^{2} - x^{3} - x$ and $0$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - x + 4 x + 2$

Step 1.1.8.2.2

Subtract $x^{3}$ from $- 2 x^{3}$ .

$- x^{4} - 3 x^{3} - x^{2} - x + 4 x + 2$

Step 1.1.8.2.3

Add $- x$ and $4 x$ .

$- x^{4} - 3 x^{3} - x^{2} + 3 x + 2$

Step 1.2

The largest exponent is the degree of the polynomial.

$4$

Step 2

Since the degree is even, the ends of the function will point in the same direction.

Even

Step 3

Identify the leading coefficient.

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Step 3.1

Simplify the polynomial, then reorder it left to right starting with the highest degree term.

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Step 3.1.1

Simplify by multiplying through.

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Step 3.1.1.1

Apply the distributive property.

$(- x - - 1) (x + 2) {(x + 1)}^{2}$

Step 3.1.1.2

Multiply $- 1$ by $- 1$ .

$(- x + 1) (x + 2) {(x + 1)}^{2}$

Step 3.1.2

Expand $(- x + 1) (x + 2)$ using the FOIL Method.

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Step 3.1.2.1

Apply the distributive property.

$(- x (x + 2) + 1 (x + 2)) {(x + 1)}^{2}$

Step 3.1.2.2

Apply the distributive property.

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

Step 3.1.2.3

Apply the distributive property.

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

Step 3.1.3

Simplify and combine like terms.

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Step 3.1.3.1

Simplify each term.

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Step 3.1.3.1.1

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

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Step 3.1.3.1.1.1

Move $x$ .

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

Step 3.1.3.1.1.2

Multiply $x$ by $x$ .

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

Step 3.1.3.1.2

Multiply $2$ by $- 1$ .

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

Step 3.1.3.1.3

Multiply $x$ by $1$ .

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

Step 3.1.3.1.4

Multiply $2$ by $1$ .

$(- x^{2} - 2 x + x + 2) {(x + 1)}^{2}$

Step 3.1.3.2

Add $- 2 x$ and $x$ .

$(- x^{2} - x + 2) {(x + 1)}^{2}$

Step 3.1.4

Rewrite ${(x + 1)}^{2}$ as $(x + 1) (x + 1)$ .

$(- x^{2} - x + 2) ((x + 1) (x + 1))$

Step 3.1.5

Expand $(x + 1) (x + 1)$ using the FOIL Method.

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Step 3.1.5.1

Apply the distributive property.

$(- x^{2} - x + 2) (x (x + 1) + 1 (x + 1))$

Step 3.1.5.2

Apply the distributive property.

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

Step 3.1.5.3

Apply the distributive property.

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

Step 3.1.6

Simplify and combine like terms.

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Step 3.1.6.1

Simplify each term.

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Step 3.1.6.1.1

Multiply $x$ by $x$ .

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

Step 3.1.6.1.2

Multiply $x$ by $1$ .

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

Step 3.1.6.1.3

Multiply $x$ by $1$ .

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

Step 3.1.6.1.4

Multiply $1$ by $1$ .

$(- x^{2} - x + 2) (x^{2} + x + x + 1)$

Step 3.1.6.2

Add $x$ and $x$ .

$(- x^{2} - x + 2) (x^{2} + 2 x + 1)$

Step 3.1.7

Expand $(- x^{2} - x + 2) (x^{2} + 2 x + 1)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 3.1.8

Simplify terms.

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Step 3.1.8.1

Simplify each term.

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Step 3.1.8.1.1

Multiply $x^{2}$ by $x^{2}$ by adding the exponents.

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Step 3.1.8.1.1.1

Move $x^{2}$ .

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

Step 3.1.8.1.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.8.1.1.3

Add $2$ and $2$ .

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

Step 3.1.8.1.2

Rewrite using the commutative property of multiplication.

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

Step 3.1.8.1.3

Multiply $x^{2}$ by $x$ by adding the exponents.

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Step 3.1.8.1.3.1

Move $x$ .

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

Step 3.1.8.1.3.2

Multiply $x$ by $x^{2}$ .

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Step 3.1.8.1.3.2.1

Raise $x$ to the power of $1$ .

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

Step 3.1.8.1.3.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.1.8.1.3.3

Add $1$ and $2$ .

$- x^{4} - 1 \cdot 2 x^{3} - x^{2} \cdot 1 - x \cdot x^{2} - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.4

Multiply $- 1$ by $2$ .

$- x^{4} - 2 x^{3} - x^{2} \cdot 1 - x \cdot x^{2} - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.5

Multiply $- 1$ by $1$ .

$- x^{4} - 2 x^{3} - x^{2} - x \cdot x^{2} - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.6

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 3.1.8.1.6.1

Move $x^{2}$ .

$- x^{4} - 2 x^{3} - x^{2} - (x^{2} x) - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.6.2

Multiply $x^{2}$ by $x$ .

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Step 3.1.8.1.6.2.1

Raise $x$ to the power of $1$ .

$- x^{4} - 2 x^{3} - x^{2} - (x^{2} x^{1}) - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.6.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$- x^{4} - 2 x^{3} - x^{2} - x^{2 + 1} - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.6.3

Add $2$ and $1$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - x (2 x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.7

Rewrite using the commutative property of multiplication.

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 1 \cdot 2 x \cdot x - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.8

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

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Step 3.1.8.1.8.1

Move $x$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 1 \cdot 2 (x \cdot x) - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.8.2

Multiply $x$ by $x$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 1 \cdot 2 x^{2} - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.9

Multiply $- 1$ by $2$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 2 x^{2} - x \cdot 1 + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.10

Multiply $- 1$ by $1$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 2 x^{2} - x + 2 x^{2} + 2 (2 x) + 2 \cdot 1$

Step 3.1.8.1.11

Multiply $2$ by $2$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 2 x^{2} - x + 2 x^{2} + 4 x + 2 \cdot 1$

Step 3.1.8.1.12

Multiply $2$ by $1$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - 2 x^{2} - x + 2 x^{2} + 4 x + 2$

Step 3.1.8.2

Simplify by adding terms.

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Step 3.1.8.2.1

Combine the opposite terms in $- x^{4} - 2 x^{3} - x^{2} - x^{3} - 2 x^{2} - x + 2 x^{2} + 4 x + 2$ .

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Step 3.1.8.2.1.1

Add $- 2 x^{2}$ and $2 x^{2}$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - x + 0 + 4 x + 2$

Step 3.1.8.2.1.2

Add $- x^{4} - 2 x^{3} - x^{2} - x^{3} - x$ and $0$ .

$- x^{4} - 2 x^{3} - x^{2} - x^{3} - x + 4 x + 2$

Step 3.1.8.2.2

Subtract $x^{3}$ from $- 2 x^{3}$ .

$- x^{4} - 3 x^{3} - x^{2} - x + 4 x + 2$

Step 3.1.8.2.3

Add $- x$ and $4 x$ .

$- x^{4} - 3 x^{3} - x^{2} + 3 x + 2$

Step 3.2

The leading term in a polynomial is the term with the highest degree.

$- x^{4}$

Step 3.3

The leading coefficient in a polynomial is the coefficient of the leading term.

$- 1$

Step 4

Since the leading coefficient is negative, the graph falls to the right.

Negative

Step 5

Use the degree of the function, as well as the sign of the leading coefficient to determine the behavior.

1. Even and Positive: Rises to the left and rises to the right.

2. Even and Negative: Falls to the left and falls to the right.

3. Odd and Positive: Falls to the left and rises to the right.

4. Odd and Negative: Rises to the left and falls to the right

Step 6

Determine the behavior.

Falls to the left and falls to the right

Step 7