Calculus Examples

$y = (x^{3} - 2) \sqrt{x^{2} + 1}$

Step 1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{2}}$ .

$\frac{d}{d x} [(x^{3} - 2) {(x^{2} + 1)}^{\frac{1}{2}}]$

Step 2

Differentiate using the Product Rule which states that $\frac{d}{d x} [f (x) g (x)]$ is $f (x) \frac{d}{d x} [g (x)] + g (x) \frac{d}{d x} [f (x)]$ where $f (x) = x^{3} - 2$ and $g (x) = {(x^{2} + 1)}^{\frac{1}{2}}$ .

$(x^{3} - 2) \frac{d}{d x} [{(x^{2} + 1)}^{\frac{1}{2}}] + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 3

Differentiate using the chain rule, which states that $\frac{d}{d x} [f (g (x))]$ is $f' (g (x)) g' (x)$ where $f (x) = x^{\frac{1}{2}}$ and $g (x) = x^{2} + 1$ .

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Step 3.1

To apply the Chain Rule, set $u$ as $x^{2} + 1$ .

$(x^{3} - 2) (\frac{d}{d u} [u^{\frac{1}{2}}] \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 3.2

Differentiate using the Power Rule which states that $\frac{d}{d u} [u^{n}]$ is $n u^{n - 1}$ where $n = \frac{1}{2}$ .

$(x^{3} - 2) (\frac{1}{2} u^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 3.3

Replace all occurrences of $u$ with $x^{2} + 1$ .

$(x^{3} - 2) (\frac{1}{2} {(x^{2} + 1)}^{\frac{1}{2} - 1} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 4

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$(x^{3} - 2) (\frac{1}{2} {(x^{2} + 1)}^{\frac{1}{2} - 1 \cdot \frac{2}{2}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 5

Combine $- 1$ and $\frac{2}{2}$ .

$(x^{3} - 2) (\frac{1}{2} {(x^{2} + 1)}^{\frac{1}{2} + \frac{- 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 6

Combine the numerators over the common denominator.

$(x^{3} - 2) (\frac{1}{2} {(x^{2} + 1)}^{\frac{1 - 1 \cdot 2}{2}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 7

Simplify the numerator.

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Step 7.1

Multiply $- 1$ by $2$ .

$(x^{3} - 2) (\frac{1}{2} {(x^{2} + 1)}^{\frac{1 - 2}{2}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 7.2

Subtract $2$ from $1$ .

$(x^{3} - 2) (\frac{1}{2} {(x^{2} + 1)}^{\frac{- 1}{2}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 8

Combine fractions.

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Step 8.1

Move the negative in front of the fraction.

$(x^{3} - 2) (\frac{1}{2} {(x^{2} + 1)}^{- \frac{1}{2}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 8.2

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

$(x^{3} - 2) (\frac{{(x^{2} + 1)}^{- \frac{1}{2}}}{2} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 8.3

Move ${(x^{2} + 1)}^{- \frac{1}{2}}$ to the denominator using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$(x^{3} - 2) (\frac{1}{2 {(x^{2} + 1)}^{\frac{1}{2}}} \frac{d}{d x} [x^{2} + 1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 9

By the Sum Rule, the derivative of $x^{2} + 1$ with respect to $x$ is $\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]$ .

$(x^{3} - 2) \frac{1}{2 {(x^{2} + 1)}^{\frac{1}{2}}} (\frac{d}{d x} [x^{2}] + \frac{d}{d x} [1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 10

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 2$ .

$(x^{3} - 2) \frac{1}{2 {(x^{2} + 1)}^{\frac{1}{2}}} (2 x + \frac{d}{d x} [1]) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 11

Since $1$ is constant with respect to $x$ , the derivative of $1$ with respect to $x$ is $0$ .

$(x^{3} - 2) \frac{1}{2 {(x^{2} + 1)}^{\frac{1}{2}}} (2 x + 0) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 12

Simplify terms.

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Step 12.1

Add $2 x$ and $0$ .

$(x^{3} - 2) \frac{1}{2 {(x^{2} + 1)}^{\frac{1}{2}}} (2 x) + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 12.2

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

$(x^{3} - 2) \frac{2}{2 {(x^{2} + 1)}^{\frac{1}{2}}} x + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 12.3

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

$(x^{3} - 2) \frac{x \cdot 2}{2 {(x^{2} + 1)}^{\frac{1}{2}}} + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 12.4

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

$(x^{3} - 2) \frac{2 \cdot x}{2 {(x^{2} + 1)}^{\frac{1}{2}}} + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 12.5

Cancel the common factor.

$(x^{3} - 2) \frac{2 x}{2 {(x^{2} + 1)}^{\frac{1}{2}}} + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 12.6

Rewrite the expression.

$(x^{3} - 2) \frac{x}{{(x^{2} + 1)}^{\frac{1}{2}}} + {(x^{2} + 1)}^{\frac{1}{2}} \frac{d}{d x} [x^{3} - 2]$

Step 13

By the Sum Rule, the derivative of $x^{3} - 2$ with respect to $x$ is $\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2]$ .

$(x^{3} - 2) \frac{x}{{(x^{2} + 1)}^{\frac{1}{2}}} + {(x^{2} + 1)}^{\frac{1}{2}} (\frac{d}{d x} [x^{3}] + \frac{d}{d x} [- 2])$

Step 14

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 3$ .

$(x^{3} - 2) \frac{x}{{(x^{2} + 1)}^{\frac{1}{2}}} + {(x^{2} + 1)}^{\frac{1}{2}} (3 x^{2} + \frac{d}{d x} [- 2])$

Step 15

Since $- 2$ is constant with respect to $x$ , the derivative of $- 2$ with respect to $x$ is $0$ .

$(x^{3} - 2) \frac{x}{{(x^{2} + 1)}^{\frac{1}{2}}} + {(x^{2} + 1)}^{\frac{1}{2}} (3 x^{2} + 0)$

Step 16

Simplify the expression.

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Step 16.1

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

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

Step 16.2

Move $3$ to the left of ${(x^{2} + 1)}^{\frac{1}{2}}$ .

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

Step 17

Simplify.

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Step 17.1

Reorder terms.

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

Step 17.2

Multiply $x^{3} - 2$ by $\frac{x}{{(x^{2} + 1)}^{\frac{1}{2}}}$ .

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

Step 17.3

To write $3 x^{2} {(x^{2} + 1)}^{\frac{1}{2}}$ as a fraction with a common denominator, multiply by $\frac{{(x^{2} + 1)}^{\frac{1}{2}}}{{(x^{2} + 1)}^{\frac{1}{2}}}$ .

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

Step 17.4

Combine the numerators over the common denominator.

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

Step 17.5

Simplify the numerator.

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Step 17.5.1

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

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Step 17.5.1.1

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

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

Step 17.5.1.2

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

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

Step 17.5.1.3

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

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

Step 17.5.2

Multiply ${(x^{2} + 1)}^{\frac{1}{2}}$ by ${(x^{2} + 1)}^{\frac{1}{2}}$ by adding the exponents.

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Step 17.5.2.1

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

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

Step 17.5.2.2

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

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

Step 17.5.2.3

Combine the numerators over the common denominator.

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

Step 17.5.2.4

Add $1$ and $1$ .

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

Step 17.5.2.5

Divide $2$ by $2$ .

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

Step 17.5.3

Simplify $3 x {(x^{2} + 1)}^{1}$ .

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

Step 17.5.4

Apply the distributive property.

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

Step 17.5.5

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

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Step 17.5.5.1

Move $x^{2}$ .

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

Step 17.5.5.2

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

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Step 17.5.5.2.1

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

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

Step 17.5.5.2.2

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

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

Step 17.5.5.3

Add $2$ and $1$ .

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

Step 17.5.6

Multiply $3$ by $1$ .

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

Step 17.5.7

Add $x^{3}$ and $3 x^{3}$ .

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

Step 17.5.8

Reorder terms.

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

Step 17.5.9

Factor $4 x^{3} + 3 x - 2$ using the rational roots test.

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Step 17.5.9.1

If a polynomial function has integer coefficients, then every rational zero will have the form $\frac{p}{q}$ where $p$ is a factor of the constant and $q$ is a factor of the leading coefficient.

$p = \pm 1, \pm 2$

$q = \pm 1, \pm 4, \pm 2$

Step 17.5.9.2

Find every combination of $\pm \frac{p}{q}$ . These are the possible roots of the polynomial function.

$\pm 1, \pm 0.25, \pm 0.5, \pm 2$

Step 17.5.9.3

Substitute $0.5$ and simplify the expression. In this case, the expression is equal to $0$ so $0.5$ is a root of the polynomial.

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Step 17.5.9.3.1

Substitute $0.5$ into the polynomial.

$4 \cdot {0.5}^{3} + 3 \cdot 0.5 - 2$

Step 17.5.9.3.2

Raise $0.5$ to the power of $3$ .

$4 \cdot 0.125 + 3 \cdot 0.5 - 2$

Step 17.5.9.3.3

Multiply $4$ by $0.125$ .

$0.5 + 3 \cdot 0.5 - 2$

Step 17.5.9.3.4

Multiply $3$ by $0.5$ .

$0.5 + 1.5 - 2$

Step 17.5.9.3.5

Add $0.5$ and $1.5$ .

$2 - 2$

Step 17.5.9.3.6

Subtract $2$ from $2$ .

$0$

Step 17.5.9.4

Since $0.5$ is a known root, divide the polynomial by $2 x - 1$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

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

Step 17.5.9.5

Divide $4 x^{3} + 3 x - 2$ by $2 x - 1$ .

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Step 17.5.9.5.1

Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of $0$ .

$2 x$

$1$

$4 x^{3}$

$0 x^{2}$

$3 x$

$2$

Step 17.5.9.5.2

Divide the highest order term in the dividend $4 x^{3}$ by the highest order term in divisor $2 x$ .

					$2 x^{2}$
	$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$

Step 17.5.9.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			+	$4 x^{3}$	-	$2 x^{2}$

Step 17.5.9.5.4

The expression needs to be subtracted from the dividend, so change all the signs in $4 x^{3} - 2 x^{2}$

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$

Step 17.5.9.5.5

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$

Step 17.5.9.5.6

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$

Step 17.5.9.5.7

Divide the highest order term in the dividend $2 x^{2}$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	+	$x$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$

Step 17.5.9.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					+	$2 x^{2}$	-	$x$

Step 17.5.9.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - x$

				$2 x^{2}$	+	$x$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$

Step 17.5.9.5.10

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	+	$x$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$

Step 17.5.9.5.11

Pull the next terms from the original dividend down into the current dividend.

				$2 x^{2}$	+	$x$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$	-	$2$

Step 17.5.9.5.12

Divide the highest order term in the dividend $4 x$ by the highest order term in divisor $2 x$ .

				$2 x^{2}$	+	$x$	+	$2$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$	-	$2$

Step 17.5.9.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$x$	+	$2$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$	-	$2$
							+	$4 x$	-	$2$

Step 17.5.9.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $4 x - 2$

				$2 x^{2}$	+	$x$	+	$2$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$	-	$2$
							-	$4 x$	+	$2$

Step 17.5.9.5.15

After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.

				$2 x^{2}$	+	$x$	+	$2$
$2 x$	-	$1$		$4 x^{3}$	+	$0 x^{2}$	+	$3 x$	-	$2$
			-	$4 x^{3}$	+	$2 x^{2}$
					+	$2 x^{2}$	+	$3 x$
					-	$2 x^{2}$	+	$x$
							+	$4 x$	-	$2$
							-	$4 x$	+	$2$
										$0$

Step 17.5.9.5.16

Since the remander is $0$ , the final answer is the quotient.

$2 x^{2} + x + 2$

Step 17.5.9.6

Write $4 x^{3} + 3 x - 2$ as a set of factors.

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

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