Precalculus Examples

$p (x) = - x^{4} - 3 x^{3} + x^{2} + 13 x + 10$

Step 1

Set $- x^{4} - 3 x^{3} + x^{2} + 13 x + 10$ equal to $0$ .

$- x^{4} - 3 x^{3} + x^{2} + 13 x + 10 = 0$

Step 2

Solve for $x$ .

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Step 2.1

Factor the left side of the equation.

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Step 2.1.1

Regroup terms.

$- x^{4} + x^{2} - 3 x^{3} + 13 x + 10 = 0$

Step 2.1.2

Factor $- x^{2}$ out of $- x^{4} + x^{2}$ .

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Step 2.1.2.1

Factor $- x^{2}$ out of $- x^{4}$ .

$- x^{2} x^{2} + x^{2} - 3 x^{3} + 13 x + 10 = 0$

Step 2.1.2.2

Factor $- x^{2}$ out of $x^{2}$ .

$- x^{2} x^{2} - x^{2} \cdot - 1 - 3 x^{3} + 13 x + 10 = 0$

Step 2.1.2.3

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

$- x^{2} (x^{2} - 1) - 3 x^{3} + 13 x + 10 = 0$

Step 2.1.3

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

$- x^{2} (x^{2} - 1^{2}) - 3 x^{3} + 13 x + 10 = 0$

Step 2.1.4

Factor.

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Step 2.1.4.1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 1$ .

$- x^{2} ((x + 1) (x - 1)) - 3 x^{3} + 13 x + 10 = 0$

Step 2.1.4.2

Remove unnecessary parentheses.

$- x^{2} (x + 1) (x - 1) - 3 x^{3} + 13 x + 10 = 0$

Step 2.1.5

Factor $- 3 x^{3} + 13 x + 10$ using the rational roots test.

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Step 2.1.5.1

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.

$p = \pm 1, \pm 10, \pm 2, \pm 5$

$q = \pm 1, \pm 3$

Step 2.1.5.2

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

$\pm 1, \pm 0.\overline{3}, \pm 10, \pm 3.\overline{3}, \pm 2, \pm 0.\overline{6}, \pm 5, \pm 1.\overline{6}$

Step 2.1.5.3

Substitute $- 1$ and simplify the expression. In this case, the expression is equal to $0$ so $- 1$ is a root of the polynomial.

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Step 2.1.5.3.1

Substitute $- 1$ into the polynomial.

$- 3 {(- 1)}^{3} + 13 \cdot - 1 + 10$

Step 2.1.5.3.2

Raise $- 1$ to the power of $3$ .

$- 3 \cdot - 1 + 13 \cdot - 1 + 10$

Step 2.1.5.3.3

Multiply $- 3$ by $- 1$ .

$3 + 13 \cdot - 1 + 10$

Step 2.1.5.3.4

Multiply $13$ by $- 1$ .

$3 - 13 + 10$

Step 2.1.5.3.5

Subtract $13$ from $3$ .

$- 10 + 10$

Step 2.1.5.3.6

Add $- 10$ and $10$ .

$0$

Step 2.1.5.4

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

$\frac{- 3 x^{3} + 13 x + 10}{x + 1}$

Step 2.1.5.5

Divide $- 3 x^{3} + 13 x + 10$ by $x + 1$ .

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Step 2.1.5.5.1

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

$x$

$1$

$3 x^{3}$

$0 x^{2}$

$13 x$

$10$

Step 2.1.5.5.2

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

				-	$3 x^{2}$
	$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$

Step 2.1.5.5.3

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			-	$3 x^{3}$	-	$3 x^{2}$

Step 2.1.5.5.4

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

			-	$3 x^{2}$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$

Step 2.1.5.5.5

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

			-	$3 x^{2}$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$

Step 2.1.5.5.6

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

			-	$3 x^{2}$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$13 x$

Step 2.1.5.5.7

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

			-	$3 x^{2}$	+	$3 x$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$13 x$

Step 2.1.5.5.8

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$	+	$3 x$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					+	$3 x^{2}$	+	$3 x$

Step 2.1.5.5.9

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

			-	$3 x^{2}$	+	$3 x$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$3 x$

Step 2.1.5.5.10

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

			-	$3 x^{2}$	+	$3 x$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$3 x$
							+	$10 x$

Step 2.1.5.5.11

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

			-	$3 x^{2}$	+	$3 x$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$3 x$
							+	$10 x$	+	$10$

Step 2.1.5.5.12

Divide the highest order term in the dividend $10 x$ by the highest order term in divisor $x$ .

			-	$3 x^{2}$	+	$3 x$	+	$10$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$3 x$
							+	$10 x$	+	$10$

Step 2.1.5.5.13

Multiply the new quotient term by the divisor.

			-	$3 x^{2}$	+	$3 x$	+	$10$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$3 x$
							+	$10 x$	+	$10$
							+	$10 x$	+	$10$

Step 2.1.5.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $10 x + 10$

			-	$3 x^{2}$	+	$3 x$	+	$10$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$3 x$
							+	$10 x$	+	$10$
							-	$10 x$	-	$10$

Step 2.1.5.5.15

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

			-	$3 x^{2}$	+	$3 x$	+	$10$
$x$	+	$1$	-	$3 x^{3}$	+	$0 x^{2}$	+	$13 x$	+	$10$
			+	$3 x^{3}$	+	$3 x^{2}$
					+	$3 x^{2}$	+	$13 x$
					-	$3 x^{2}$	-	$3 x$
							+	$10 x$	+	$10$
							-	$10 x$	-	$10$
										$0$

Step 2.1.5.5.16

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

$- 3 x^{2} + 3 x + 10$

Step 2.1.5.6

Write $- 3 x^{3} + 13 x + 10$ as a set of factors.

$- x^{2} (x + 1) (x - 1) + (x + 1) (- 3 x^{2} + 3 x + 10) = 0$

Step 2.1.6

Factor $x + 1$ out of $- x^{2} (x + 1) (x - 1) + (x + 1) (- 3 x^{2} + 3 x + 10)$ .

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Step 2.1.6.1

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

$(x + 1) (- x^{2} (x - 1)) + (x + 1) (- 3 x^{2} + 3 x + 10) = 0$

Step 2.1.6.2

Factor $x + 1$ out of $(x + 1) (- x^{2} (x - 1)) + (x + 1) (- 3 x^{2} + 3 x + 10)$ .

$(x + 1) (- x^{2} (x - 1) - 3 x^{2} + 3 x + 10) = 0$

Step 2.1.7

Apply the distributive property.

$(x + 1) (- x^{2} x - x^{2} \cdot - 1 - 3 x^{2} + 3 x + 10) = 0$

Step 2.1.8

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

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Step 2.1.8.1

Move $x$ .

$(x + 1) (- (x \cdot x^{2}) - x^{2} \cdot - 1 - 3 x^{2} + 3 x + 10) = 0$

Step 2.1.8.2

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

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Step 2.1.8.2.1

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

$(x + 1) (- (x \cdot x^{2}) - x^{2} \cdot - 1 - 3 x^{2} + 3 x + 10) = 0$

Step 2.1.8.2.2

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

$(x + 1) (- x^{1 + 2} - x^{2} \cdot - 1 - 3 x^{2} + 3 x + 10) = 0$

Step 2.1.8.3

Add $1$ and $2$ .

$(x + 1) (- x^{3} - x^{2} \cdot - 1 - 3 x^{2} + 3 x + 10) = 0$

Step 2.1.9

Multiply $- x^{2} \cdot - 1$ .

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Step 2.1.9.1

Multiply $- 1$ by $- 1$ .

$(x + 1) (- x^{3} + 1 x^{2} - 3 x^{2} + 3 x + 10) = 0$

Step 2.1.9.2

Multiply $x^{2}$ by $1$ .

$(x + 1) (- x^{3} + x^{2} - 3 x^{2} + 3 x + 10) = 0$

Step 2.1.10

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

$(x + 1) (- x^{3} - 2 x^{2} + 3 x + 10) = 0$

Step 2.1.11

Factor.

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Step 2.1.11.1

Factor $- x^{3} - 2 x^{2} + 3 x + 10$ using the rational roots test.

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Step 2.1.11.1.1

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.

$p = \pm 1, \pm 10, \pm 2, \pm 5$

$q = \pm 1$

Step 2.1.11.1.2

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

$\pm 1, \pm 10, \pm 2, \pm 5$

Step 2.1.11.1.3

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 2.1.11.1.3.1

Substitute $2$ into the polynomial.

$- 2^{3} - 2 \cdot 2^{2} + 3 \cdot 2 + 10$

Step 2.1.11.1.3.2

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

$- 1 \cdot 8 - 2 \cdot 2^{2} + 3 \cdot 2 + 10$

Step 2.1.11.1.3.3

Multiply $- 1$ by $8$ .

$- 8 - 2 \cdot 2^{2} + 3 \cdot 2 + 10$

Step 2.1.11.1.3.4

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

$- 8 - 2 \cdot 4 + 3 \cdot 2 + 10$

Step 2.1.11.1.3.5

Multiply $- 2$ by $4$ .

$- 8 - 8 + 3 \cdot 2 + 10$

Step 2.1.11.1.3.6

Subtract $8$ from $- 8$ .

$- 16 + 3 \cdot 2 + 10$

Step 2.1.11.1.3.7

Multiply $3$ by $2$ .

$- 16 + 6 + 10$

Step 2.1.11.1.3.8

Add $- 16$ and $6$ .

$- 10 + 10$

Step 2.1.11.1.3.9

Add $- 10$ and $10$ .

$0$

Step 2.1.11.1.4

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.

$\frac{- x^{3} - 2 x^{2} + 3 x + 10}{x - 2}$

Step 2.1.11.1.5

Divide $- x^{3} - 2 x^{2} + 3 x + 10$ by $x - 2$ .

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Step 2.1.11.1.5.1

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

$x$

$2$

$x^{3}$

$2 x^{2}$

$3 x$

$10$

Step 2.1.11.1.5.2

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

				-	$x^{2}$
	$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$

Step 2.1.11.1.5.3

Multiply the new quotient term by the divisor.

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			-	$x^{3}$	+	$2 x^{2}$

Step 2.1.11.1.5.4

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

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$

Step 2.1.11.1.5.5

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

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$

Step 2.1.11.1.5.6

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

			-	$x^{2}$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	+	$3 x$

Step 2.1.11.1.5.7

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

			-	$x^{2}$	-	$4 x$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	+	$3 x$

Step 2.1.11.1.5.8

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$4 x$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	+	$3 x$
					-	$4 x^{2}$	+	$8 x$

Step 2.1.11.1.5.9

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

			-	$x^{2}$	-	$4 x$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	+	$3 x$
					+	$4 x^{2}$	-	$8 x$

Step 2.1.11.1.5.10

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

			-	$x^{2}$	-	$4 x$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	+	$3 x$
					+	$4 x^{2}$	-	$8 x$
							-	$5 x$

Step 2.1.11.1.5.11

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

			-	$x^{2}$	-	$4 x$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	+	$3 x$
					+	$4 x^{2}$	-	$8 x$
							-	$5 x$	+	$10$

Step 2.1.11.1.5.12

Divide the highest order term in the dividend $- 5 x$ by the highest order term in divisor $x$ .

			-	$x^{2}$	-	$4 x$	-	$5$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	+	$3 x$
					+	$4 x^{2}$	-	$8 x$
							-	$5 x$	+	$10$

Step 2.1.11.1.5.13

Multiply the new quotient term by the divisor.

			-	$x^{2}$	-	$4 x$	-	$5$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	+	$3 x$
					+	$4 x^{2}$	-	$8 x$
							-	$5 x$	+	$10$
							-	$5 x$	+	$10$

Step 2.1.11.1.5.14

The expression needs to be subtracted from the dividend, so change all the signs in $- 5 x + 10$

			-	$x^{2}$	-	$4 x$	-	$5$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	+	$3 x$
					+	$4 x^{2}$	-	$8 x$
							-	$5 x$	+	$10$
							+	$5 x$	-	$10$

Step 2.1.11.1.5.15

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

			-	$x^{2}$	-	$4 x$	-	$5$
$x$	-	$2$	-	$x^{3}$	-	$2 x^{2}$	+	$3 x$	+	$10$
			+	$x^{3}$	-	$2 x^{2}$
					-	$4 x^{2}$	+	$3 x$
					+	$4 x^{2}$	-	$8 x$
							-	$5 x$	+	$10$
							+	$5 x$	-	$10$
										$0$

Step 2.1.11.1.5.16

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

$- x^{2} - 4 x - 5$

Step 2.1.11.1.6

Write $- x^{3} - 2 x^{2} + 3 x + 10$ as a set of factors.

$(x + 1) ((x - 2) (- x^{2} - 4 x - 5)) = 0$

Step 2.1.11.2

Remove unnecessary parentheses.

$(x + 1) (x - 2) (- x^{2} - 4 x - 5) = 0$

Step 2.2

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

$x + 1 = 0$

$x - 2 = 0$

$- x^{2} - 4 x - 5 = 0$

Step 2.3

Set $x + 1$ equal to $0$ and solve for $x$ .

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Step 2.3.1

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

$x + 1 = 0$

Step 2.3.2

Subtract $1$ from both sides of the equation.

$x = - 1$

Step 2.4

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

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Step 2.4.1

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

$x - 2 = 0$

Step 2.4.2

Add $2$ to both sides of the equation.

$x = 2$

Step 2.5

Set $- x^{2} - 4 x - 5$ equal to $0$ and solve for $x$ .

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Step 2.5.1

Set $- x^{2} - 4 x - 5$ equal to $0$ .

$- x^{2} - 4 x - 5 = 0$

Step 2.5.2

Solve $- x^{2} - 4 x - 5 = 0$ for $x$ .

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Step 2.5.2.1

Use the quadratic formula to find the solutions.

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

Step 2.5.2.2

Substitute the values $a = - 1$ , $b = - 4$ , and $c = - 5$ into the quadratic formula and solve for $x$ .

$\frac{4 \pm \sqrt{{(- 4)}^{2} - 4 \cdot (- 1 \cdot - 5)}}{2 \cdot - 1}$

Step 2.5.2.3

Simplify.

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Step 2.5.2.3.1

Simplify the numerator.

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Step 2.5.2.3.1.1

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

$x = \frac{4 \pm \sqrt{16 - 4 \cdot - 1 \cdot - 5}}{2 \cdot - 1}$

Step 2.5.2.3.1.2

Multiply $- 4 \cdot - 1 \cdot - 5$ .

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Step 2.5.2.3.1.2.1

Multiply $- 4$ by $- 1$ .

$x = \frac{4 \pm \sqrt{16 + 4 \cdot - 5}}{2 \cdot - 1}$

Step 2.5.2.3.1.2.2

Multiply $4$ by $- 5$ .

$x = \frac{4 \pm \sqrt{16 - 20}}{2 \cdot - 1}$

Step 2.5.2.3.1.3

Subtract $20$ from $16$ .

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

Step 2.5.2.3.1.4

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

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

Step 2.5.2.3.1.5

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

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

Step 2.5.2.3.1.6

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

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

Step 2.5.2.3.1.7

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

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

Step 2.5.2.3.1.8

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

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

Step 2.5.2.3.1.9

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

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

Step 2.5.2.3.2

Multiply $2$ by $- 1$ .

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

Step 2.5.2.3.3

Simplify $\frac{4 \pm 2 i}{- 2}$ .

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

Step 2.5.2.3.4

Move the negative one from the denominator of $\frac{2 \pm i}{- 1}$ .

$x = - 1 \cdot (2 \pm i)$

Step 2.5.2.3.5

Rewrite $- 1 \cdot (2 \pm i)$ as $- (2 \pm i)$ .

$x = - (2 \pm i)$

Step 2.5.2.4

The final answer is the combination of both solutions.

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

Step 2.6

The final solution is all the values that make $(x + 1) (x - 2) (- x^{2} - 4 x - 5) = 0$ true.

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

Step 3