Precalculus Examples

$x^{4} + x^{3} - 6 x^{2} - 14 x - 12 = 0$

Step 1

Factor the left side of the equation.

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Step 1.1

Factor $x^{4} + x^{3} - 6 x^{2} - 14 x - 12$ using the rational roots test.

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Step 1.1.1

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.

$p = \pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

$q = \pm 1$

Step 1.1.2

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

$\pm 1, \pm 12, \pm 2, \pm 6, \pm 3, \pm 4$

Step 1.1.3

Substitute $- 2$ and simplify the expression. In this case, the expression is equal to $0$ so $- 2$ is a root of the polynomial.

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Step 1.1.3.1

Substitute $- 2$ into the polynomial.

${(- 2)}^{4} + {(- 2)}^{3} - 6 {(- 2)}^{2} - 14 \cdot - 2 - 12$

Step 1.1.3.2

Raise $- 2$ to the power of $4$ .

$16 + {(- 2)}^{3} - 6 {(- 2)}^{2} - 14 \cdot - 2 - 12$

Step 1.1.3.3

Raise $- 2$ to the power of $3$ .

$16 - 8 - 6 {(- 2)}^{2} - 14 \cdot - 2 - 12$

Step 1.1.3.4

Subtract $8$ from $16$ .

$8 - 6 {(- 2)}^{2} - 14 \cdot - 2 - 12$

Step 1.1.3.5

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

$8 - 6 \cdot 4 - 14 \cdot - 2 - 12$

Step 1.1.3.6

Multiply $- 6$ by $4$ .

$8 - 24 - 14 \cdot - 2 - 12$

Step 1.1.3.7

Subtract $24$ from $8$ .

$- 16 - 14 \cdot - 2 - 12$

Step 1.1.3.8

Multiply $- 14$ by $- 2$ .

$- 16 + 28 - 12$

Step 1.1.3.9

Add $- 16$ and $28$ .

$12 - 12$

Step 1.1.3.10

Subtract $12$ from $12$ .

$0$

Step 1.1.4

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

$\frac{x^{4} + x^{3} - 6 x^{2} - 14 x - 12}{x + 2}$

Step 1.1.5

Divide $x^{4} + x^{3} - 6 x^{2} - 14 x - 12$ by $x + 2$ .

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Step 1.1.5.1

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

$x$

$2$

$x^{4}$

$x^{3}$

$6 x^{2}$

$14 x$

$12$

Step 1.1.5.2

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

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$6 x^{2}$

$14 x$

$12$

Step 1.1.5.3

Multiply the new quotient term by the divisor.

$x^{3}$

$x$

$2$

$x^{4}$

$x^{3}$

$6 x^{2}$

$14 x$

$12$

$x^{4}$

$2 x^{3}$

Step 1.1.5.4

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

				$x^{3}$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$12$
			-	$x^{4}$	-	$2 x^{3}$

Step 1.1.5.5

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

				$x^{3}$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$12$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$

Step 1.1.5.6

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

				$x^{3}$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$12$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$6 x^{2}$

Step 1.1.5.7

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

				$x^{3}$	-	$x^{2}$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$12$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$6 x^{2}$

Step 1.1.5.8

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$x^{2}$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$12$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$6 x^{2}$
					-	$x^{3}$	-	$2 x^{2}$

Step 1.1.5.9

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

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$6 x^{2}$

$14 x$

$12$

$x^{4}$

$2 x^{3}$

$x^{3}$

$6 x^{2}$

$x^{3}$

$2 x^{2}$

Step 1.1.5.10

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

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$6 x^{2}$

$14 x$

$12$

$x^{4}$

$2 x^{3}$

$x^{3}$

$6 x^{2}$

$x^{3}$

$2 x^{2}$

$4 x^{2}$

Step 1.1.5.11

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

$x^{3}$

$x^{2}$

$x$

$2$

$x^{4}$

$x^{3}$

$6 x^{2}$

$14 x$

$12$

$x^{4}$

$2 x^{3}$

$x^{3}$

$6 x^{2}$

$x^{3}$

$2 x^{2}$

$4 x^{2}$

$14 x$

Step 1.1.5.12

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

$x^{3}$

$x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$x^{3}$

$6 x^{2}$

$14 x$

$12$

$x^{4}$

$2 x^{3}$

$x^{3}$

$6 x^{2}$

$x^{3}$

$2 x^{2}$

$4 x^{2}$

$14 x$

Step 1.1.5.13

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$x^{2}$	-	$4 x$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$12$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$6 x^{2}$
					+	$x^{3}$	+	$2 x^{2}$
							-	$4 x^{2}$	-	$14 x$
							-	$4 x^{2}$	-	$8 x$

Step 1.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 4 x^{2} - 8 x$

$x^{3}$

$x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$x^{3}$

$6 x^{2}$

$14 x$

$12$

$x^{4}$

$2 x^{3}$

$x^{3}$

$6 x^{2}$

$x^{3}$

$2 x^{2}$

$4 x^{2}$

$14 x$

$4 x^{2}$

$8 x$

Step 1.1.5.15

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

$x^{3}$

$x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$x^{3}$

$6 x^{2}$

$14 x$

$12$

$x^{4}$

$2 x^{3}$

$x^{3}$

$6 x^{2}$

$x^{3}$

$2 x^{2}$

$4 x^{2}$

$14 x$

$4 x^{2}$

$8 x$

$6 x$

Step 1.1.5.16

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

$x^{3}$

$x^{2}$

$4 x$

$x$

$2$

$x^{4}$

$x^{3}$

$6 x^{2}$

$14 x$

$12$

$x^{4}$

$2 x^{3}$

$x^{3}$

$6 x^{2}$

$x^{3}$

$2 x^{2}$

$4 x^{2}$

$14 x$

$4 x^{2}$

$8 x$

$6 x$

$12$

Step 1.1.5.17

Divide the highest order term in the dividend $- 6 x$ by the highest order term in divisor $x$ .

				$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$12$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$6 x^{2}$
					+	$x^{3}$	+	$2 x^{2}$
							-	$4 x^{2}$	-	$14 x$
							+	$4 x^{2}$	+	$8 x$
									-	$6 x$	-	$12$

Step 1.1.5.18

Multiply the new quotient term by the divisor.

				$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$12$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$6 x^{2}$
					+	$x^{3}$	+	$2 x^{2}$
							-	$4 x^{2}$	-	$14 x$
							+	$4 x^{2}$	+	$8 x$
									-	$6 x$	-	$12$
									-	$6 x$	-	$12$

Step 1.1.5.19

The expression needs to be subtracted from the dividend, so change all the signs in $- 6 x - 12$

				$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$12$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$6 x^{2}$
					+	$x^{3}$	+	$2 x^{2}$
							-	$4 x^{2}$	-	$14 x$
							+	$4 x^{2}$	+	$8 x$
									-	$6 x$	-	$12$
									+	$6 x$	+	$12$

Step 1.1.5.20

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

				$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
$x$	+	$2$		$x^{4}$	+	$x^{3}$	-	$6 x^{2}$	-	$14 x$	-	$12$
			-	$x^{4}$	-	$2 x^{3}$
					-	$x^{3}$	-	$6 x^{2}$
					+	$x^{3}$	+	$2 x^{2}$
							-	$4 x^{2}$	-	$14 x$
							+	$4 x^{2}$	+	$8 x$
									-	$6 x$	-	$12$
									+	$6 x$	+	$12$
												$0$

Step 1.1.5.21

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

$x^{3} - x^{2} - 4 x - 6$

Step 1.1.6

Write $x^{4} + x^{3} - 6 x^{2} - 14 x - 12$ as a set of factors.

$(x + 2) (x^{3} - x^{2} - 4 x - 6) = 0$

Step 1.2

Factor $x^{3} - x^{2} - 4 x - 6$ using the rational roots test.

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Step 1.2.1

Factor $x^{3} - x^{2} - 4 x - 6$ using the rational roots test.

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Step 1.2.1.1

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.

$p = \pm 1, \pm 6, \pm 2, \pm 3$

$q = \pm 1$

Step 1.2.1.2

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

$\pm 1, \pm 6, \pm 2, \pm 3$

Step 1.2.1.3

Substitute $3$ and simplify the expression. In this case, the expression is equal to $0$ so $3$ is a root of the polynomial.

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Step 1.2.1.3.1

Substitute $3$ into the polynomial.

$3^{3} - 3^{2} - 4 \cdot 3 - 6$

Step 1.2.1.3.2

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

$27 - 3^{2} - 4 \cdot 3 - 6$

Step 1.2.1.3.3

Raise $3$ to the power of $2$ .

$27 - 1 \cdot 9 - 4 \cdot 3 - 6$

Step 1.2.1.3.4

Multiply $- 1$ by $9$ .

$27 - 9 - 4 \cdot 3 - 6$

Step 1.2.1.3.5

Subtract $9$ from $27$ .

$18 - 4 \cdot 3 - 6$

Step 1.2.1.3.6

Multiply $- 4$ by $3$ .

$18 - 12 - 6$

Step 1.2.1.3.7

Subtract $12$ from $18$ .

$6 - 6$

Step 1.2.1.3.8

Subtract $6$ from $6$ .

$0$

Step 1.2.1.4

Since $3$ is a known root, divide the polynomial by $x - 3$ to find the quotient polynomial. This polynomial can then be used to find the remaining roots.

$\frac{x^{3} - x^{2} - 4 x - 6}{x - 3}$

Step 1.2.1.5

Divide $x^{3} - x^{2} - 4 x - 6$ by $x - 3$ .

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Step 1.2.1.5.1

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

$x$

$3$

$x^{3}$

$x^{2}$

$4 x$

$6$

Step 1.2.1.5.2

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

					$x^{2}$
	$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$

Step 1.2.1.5.3

Multiply the new quotient term by the divisor.

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			+	$x^{3}$	-	$3 x^{2}$

Step 1.2.1.5.4

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$

Step 1.2.1.5.5

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$

Step 1.2.1.5.6

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

				$x^{2}$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$4 x$

Step 1.2.1.5.7

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

				$x^{2}$	+	$2 x$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$4 x$

Step 1.2.1.5.8

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$4 x$
					+	$2 x^{2}$	-	$6 x$

Step 1.2.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $2 x^{2} - 6 x$

				$x^{2}$	+	$2 x$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$4 x$
					-	$2 x^{2}$	+	$6 x$

Step 1.2.1.5.10

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

				$x^{2}$	+	$2 x$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$4 x$
					-	$2 x^{2}$	+	$6 x$
							+	$2 x$

Step 1.2.1.5.11

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

				$x^{2}$	+	$2 x$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$4 x$
					-	$2 x^{2}$	+	$6 x$
							+	$2 x$	-	$6$

Step 1.2.1.5.12

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

				$x^{2}$	+	$2 x$	+	$2$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$4 x$
					-	$2 x^{2}$	+	$6 x$
							+	$2 x$	-	$6$

Step 1.2.1.5.13

Multiply the new quotient term by the divisor.

				$x^{2}$	+	$2 x$	+	$2$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$4 x$
					-	$2 x^{2}$	+	$6 x$
							+	$2 x$	-	$6$
							+	$2 x$	-	$6$

Step 1.2.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $2 x - 6$

				$x^{2}$	+	$2 x$	+	$2$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$4 x$
					-	$2 x^{2}$	+	$6 x$
							+	$2 x$	-	$6$
							-	$2 x$	+	$6$

Step 1.2.1.5.15

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

				$x^{2}$	+	$2 x$	+	$2$
$x$	-	$3$		$x^{3}$	-	$x^{2}$	-	$4 x$	-	$6$
			-	$x^{3}$	+	$3 x^{2}$
					+	$2 x^{2}$	-	$4 x$
					-	$2 x^{2}$	+	$6 x$
							+	$2 x$	-	$6$
							-	$2 x$	+	$6$
										$0$

Step 1.2.1.5.16

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

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

Step 1.2.1.6

Write $x^{3} - x^{2} - 4 x - 6$ as a set of factors.

$(x + 2) ((x - 3) (x^{2} + 2 x + 2)) = 0$

Step 1.2.2

Remove unnecessary parentheses.

$(x + 2) (x - 3) (x^{2} + 2 x + 2) = 0$

Step 2

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

$x + 2 = 0$

$x - 3 = 0$

$x^{2} + 2 x + 2 = 0$

Step 3

Set $x + 2$ equal to $0$ and solve for $x$ .

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Step 3.1

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Step 3.2

Subtract $2$ from both sides of the equation.

$x = - 2$

Step 4

Set $x - 3$ equal to $0$ and solve for $x$ .

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Step 4.1

Set $x - 3$ equal to $0$ .

$x - 3 = 0$

Step 4.2

Add $3$ to both sides of the equation.

$x = 3$

Step 5

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

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Step 5.1

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

$x^{2} + 2 x + 2 = 0$

Step 5.2

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

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Step 5.2.1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 5.2.2

Substitute the values $a = 1$ , $b = 2$ , and $c = 2$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (1 \cdot 2)}}{2 \cdot 1}$

Step 5.2.3

Simplify.

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Step 5.2.3.1

Simplify the numerator.

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Step 5.2.3.1.1

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

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 1 \cdot 2}}{2 \cdot 1}$

Step 5.2.3.1.2

Multiply $- 4 \cdot 1 \cdot 2$ .

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Step 5.2.3.1.2.1

Multiply $- 4$ by $1$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot 2}}{2 \cdot 1}$

Step 5.2.3.1.2.2

Multiply $- 4$ by $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 8}}{2 \cdot 1}$

Step 5.2.3.1.3

Subtract $8$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 4}}{2 \cdot 1}$

Step 5.2.3.1.4

Rewrite $- 4$ as $- 1 (4)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 4}}{2 \cdot 1}$

Step 5.2.3.1.5

Rewrite $\sqrt{- 1 (4)}$ as $\sqrt{- 1} \cdot \sqrt{4}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{4}}{2 \cdot 1}$

Step 5.2.3.1.6

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{4}}{2 \cdot 1}$

Step 5.2.3.1.7

Rewrite $4$ as $2^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{2^{2}}}{2 \cdot 1}$

Step 5.2.3.1.8

Pull terms out from under the radical, assuming positive real numbers.

$x = \frac{- 2 \pm i \cdot 2}{2 \cdot 1}$

Step 5.2.3.1.9

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

$x = \frac{- 2 \pm 2 i}{2 \cdot 1}$

Step 5.2.3.2

Multiply $2$ by $1$ .

$x = \frac{- 2 \pm 2 i}{2}$

Step 5.2.3.3

Simplify $\frac{- 2 \pm 2 i}{2}$ .

$x = - 1 \pm i$

Step 5.2.4

The final answer is the combination of both solutions.

$x = - 1 + i, - 1 - i$

Step 6

The final solution is all the values that make $(x + 2) (x - 3) (x^{2} + 2 x + 2) = 0$ true.

$x = - 2, 3, - 1 + i, - 1 - i$