Algebra Examples

$(x + 4) (a x^{2} + b x + c) = 2 x^{3} + 9 x^{2} + 3 x - 4$

Step 1

Divide each term in $(x + 4) (a x^{2} + b x + c) = 2 x^{3} + 9 x^{2} + 3 x - 4$ by $x + 4$ and simplify.

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Step 1.1

Divide each term in $(x + 4) (a x^{2} + b x + c) = 2 x^{3} + 9 x^{2} + 3 x - 4$ by $x + 4$ .

$\frac{(x + 4) (a x^{2} + b x + c)}{x + 4} = \frac{2 x^{3}}{x + 4} + \frac{9 x^{2}}{x + 4} + \frac{3 x}{x + 4} + \frac{- 4}{x + 4}$

Step 1.2

Simplify the left side.

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Step 1.2.1

Cancel the common factor of $x + 4$ .

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Step 1.2.1.1

Cancel the common factor.

$\frac{(x + 4) (a x^{2} + b x + c)}{x + 4} = \frac{2 x^{3}}{x + 4} + \frac{9 x^{2}}{x + 4} + \frac{3 x}{x + 4} + \frac{- 4}{x + 4}$

Step 1.2.1.2

Divide $a x^{2} + b x + c$ by $1$ .

$a x^{2} + b x + c = \frac{2 x^{3}}{x + 4} + \frac{9 x^{2}}{x + 4} + \frac{3 x}{x + 4} + \frac{- 4}{x + 4}$

Step 1.3

Simplify the right side.

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Step 1.3.1

Combine the numerators over the common denominator.

$a x^{2} + b x + c = \frac{2 x^{3} + 9 x^{2} + 3 x - 4}{x + 4}$

Step 2

Move all terms not containing $a$ to the right side of the equation.

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Step 2.1

Subtract $b x$ from both sides of the equation.

$a x^{2} + c = \frac{2 x^{3} + 9 x^{2} + 3 x - 4}{x + 4} - b x$

Step 2.2

Subtract $c$ from both sides of the equation.

$a x^{2} = \frac{2 x^{3} + 9 x^{2} + 3 x - 4}{x + 4} - b x - c$

Step 2.3

Simplify each term.

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Step 2.3.1

Split the fraction $\frac{2 x^{3} + 9 x^{2} + 3 x - 4}{x + 4}$ into two fractions.

$a x^{2} = \frac{2 x^{3} + 9 x^{2} + 3 x}{x + 4} + \frac{- 4}{x + 4} - b x - c$

Step 2.3.2

Simplify each term.

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Step 2.3.2.1

Factor $x$ out of $2 x^{3} + 9 x^{2} + 3 x$ .

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Step 2.3.2.1.1

Factor $x$ out of $2 x^{3}$ .

$a x^{2} = \frac{x (2 x^{2}) + 9 x^{2} + 3 x}{x + 4} + \frac{- 4}{x + 4} - b x - c$

Step 2.3.2.1.2

Factor $x$ out of $9 x^{2}$ .

$a x^{2} = \frac{x (2 x^{2}) + x (9 x) + 3 x}{x + 4} + \frac{- 4}{x + 4} - b x - c$

Step 2.3.2.1.3

Factor $x$ out of $3 x$ .

$a x^{2} = \frac{x (2 x^{2}) + x (9 x) + x \cdot 3}{x + 4} + \frac{- 4}{x + 4} - b x - c$

Step 2.3.2.1.4

Factor $x$ out of $x (2 x^{2}) + x (9 x)$ .

$a x^{2} = \frac{x (2 x^{2} + 9 x) + x \cdot 3}{x + 4} + \frac{- 4}{x + 4} - b x - c$

Step 2.3.2.1.5

Factor $x$ out of $x (2 x^{2} + 9 x) + x \cdot 3$ .

$a x^{2} = \frac{x (2 x^{2} + 9 x + 3)}{x + 4} + \frac{- 4}{x + 4} - b x - c$

Step 2.3.2.2

Move the negative in front of the fraction.

$a x^{2} = \frac{x (2 x^{2} + 9 x + 3)}{x + 4} - \frac{4}{x + 4} - b x - c$

Step 3

Divide each term in $a x^{2} = \frac{x (2 x^{2} + 9 x + 3)}{x + 4} - \frac{4}{x + 4} - b x - c$ by $x^{2}$ and simplify.

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Step 3.1

Divide each term in $a x^{2} = \frac{x (2 x^{2} + 9 x + 3)}{x + 4} - \frac{4}{x + 4} - b x - c$ by $x^{2}$ .

$\frac{a x^{2}}{x^{2}} = \frac{\frac{x (2 x^{2} + 9 x + 3)}{x + 4}}{x^{2}} + \frac{- \frac{4}{x + 4}}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 3.2

Simplify the left side.

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Step 3.2.1

Cancel the common factor of $x^{2}$ .

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Step 3.2.1.1

Cancel the common factor.

$\frac{a x^{2}}{x^{2}} = \frac{\frac{x (2 x^{2} + 9 x + 3)}{x + 4}}{x^{2}} + \frac{- \frac{4}{x + 4}}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 3.2.1.2

Divide $a$ by $1$ .

$a = \frac{\frac{x (2 x^{2} + 9 x + 3)}{x + 4}}{x^{2}} + \frac{- \frac{4}{x + 4}}{x^{2}} + \frac{- b x}{x^{2}} + \frac{- c}{x^{2}}$

Step 3.3

Simplify the right side.

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Step 3.3.1

Simplify terms.

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Step 3.3.1.1

Combine the numerators over the common denominator.

$a = \frac{\frac{x (2 x^{2} + 9 x + 3)}{x + 4} - \frac{4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.1.2

Combine the numerators over the common denominator.

$a = \frac{\frac{x (2 x^{2} + 9 x + 3) - 4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2

Simplify each term.

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Step 3.3.2.1

Simplify the numerator.

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Step 3.3.2.1.1

Apply the distributive property.

$a = \frac{\frac{x (2 x^{2}) + x (9 x) + x \cdot 3 - 4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.2

Simplify.

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Step 3.3.2.1.2.1

Rewrite using the commutative property of multiplication.

$a = \frac{\frac{2 x \cdot x^{2} + x (9 x) + x \cdot 3 - 4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.2.2

Rewrite using the commutative property of multiplication.

$a = \frac{\frac{2 x \cdot x^{2} + 9 x \cdot x + x \cdot 3 - 4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.2.3

Move $3$ to the left of $x$ .

$a = \frac{\frac{2 x \cdot x^{2} + 9 x \cdot x + 3 \cdot x - 4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.3

Simplify each term.

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Step 3.3.2.1.3.1

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

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Step 3.3.2.1.3.1.1

Move $x^{2}$ .

$a = \frac{\frac{2 (x^{2} x) + 9 x \cdot x + 3 \cdot x - 4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.3.1.2

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

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Step 3.3.2.1.3.1.2.1

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

$a = \frac{\frac{2 (x^{2} x^{1}) + 9 x \cdot x + 3 \cdot x - 4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.3.1.2.2

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

$a = \frac{\frac{2 x^{2 + 1} + 9 x \cdot x + 3 \cdot x - 4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.3.1.3

Add $2$ and $1$ .

$a = \frac{\frac{2 x^{3} + 9 x \cdot x + 3 \cdot x - 4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.3.2

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

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Step 3.3.2.1.3.2.1

Move $x$ .

$a = \frac{\frac{2 x^{3} + 9 (x \cdot x) + 3 \cdot x - 4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.3.2.2

Multiply $x$ by $x$ .

$a = \frac{\frac{2 x^{3} + 9 x^{2} + 3 \cdot x - 4}{x + 4} - b x - c}{x^{2}}$

$a = \frac{\frac{2 x^{3} + 9 x^{2} + 3 x - 4}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.4

Rewrite $2 x^{3} + 9 x^{2} + 3 x - 4$ in a factored form.

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Step 3.3.2.1.4.1

Factor $2 x^{3} + 9 x^{2} + 3 x - 4$ using the rational roots test.

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Step 3.3.2.1.4.1.1

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.

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

$q = \pm 1, \pm 2$

Step 3.3.2.1.4.1.2

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

$\pm 1, \pm 0.5, \pm 4, \pm 2$

Step 3.3.2.1.4.1.3

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 3.3.2.1.4.1.3.1

Substitute $- 1$ into the polynomial.

$2 {(- 1)}^{3} + 9 {(- 1)}^{2} + 3 \cdot - 1 - 4$

Step 3.3.2.1.4.1.3.2

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

$2 \cdot - 1 + 9 {(- 1)}^{2} + 3 \cdot - 1 - 4$

Step 3.3.2.1.4.1.3.3

Multiply $2$ by $- 1$ .

$- 2 + 9 {(- 1)}^{2} + 3 \cdot - 1 - 4$

Step 3.3.2.1.4.1.3.4

Raise $- 1$ to the power of $2$ .

$- 2 + 9 \cdot 1 + 3 \cdot - 1 - 4$

Step 3.3.2.1.4.1.3.5

Multiply $9$ by $1$ .

$- 2 + 9 + 3 \cdot - 1 - 4$

Step 3.3.2.1.4.1.3.6

Add $- 2$ and $9$ .

$7 + 3 \cdot - 1 - 4$

Step 3.3.2.1.4.1.3.7

Multiply $3$ by $- 1$ .

$7 - 3 - 4$

Step 3.3.2.1.4.1.3.8

Subtract $3$ from $7$ .

$4 - 4$

Step 3.3.2.1.4.1.3.9

Subtract $4$ from $4$ .

$0$

Step 3.3.2.1.4.1.4

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.

$\frac{2 x^{3} + 9 x^{2} + 3 x - 4}{x + 1}$

Step 3.3.2.1.4.1.5

Divide $2 x^{3} + 9 x^{2} + 3 x - 4$ by $x + 1$ .

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Step 3.3.2.1.4.1.5.1

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

$x$

$1$

$2 x^{3}$

$9 x^{2}$

$3 x$

$4$

Step 3.3.2.1.4.1.5.2

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

					$2 x^{2}$
	$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$

Step 3.3.2.1.4.1.5.3

Multiply the new quotient term by the divisor.

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			+	$2 x^{3}$	+	$2 x^{2}$

Step 3.3.2.1.4.1.5.4

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$

Step 3.3.2.1.4.1.5.5

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$

Step 3.3.2.1.4.1.5.6

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

				$2 x^{2}$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	+	$3 x$

Step 3.3.2.1.4.1.5.7

Divide the highest order term in the dividend $7 x^{2}$ by the highest order term in divisor $x$ .

				$2 x^{2}$	+	$7 x$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	+	$3 x$

Step 3.3.2.1.4.1.5.8

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$7 x$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	+	$3 x$
					+	$7 x^{2}$	+	$7 x$

Step 3.3.2.1.4.1.5.9

The expression needs to be subtracted from the dividend, so change all the signs in $7 x^{2} + 7 x$

				$2 x^{2}$	+	$7 x$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	+	$3 x$
					-	$7 x^{2}$	-	$7 x$

Step 3.3.2.1.4.1.5.10

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

				$2 x^{2}$	+	$7 x$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	+	$3 x$
					-	$7 x^{2}$	-	$7 x$
							-	$4 x$

Step 3.3.2.1.4.1.5.11

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

				$2 x^{2}$	+	$7 x$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	+	$3 x$
					-	$7 x^{2}$	-	$7 x$
							-	$4 x$	-	$4$

Step 3.3.2.1.4.1.5.12

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

				$2 x^{2}$	+	$7 x$	-	$4$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	+	$3 x$
					-	$7 x^{2}$	-	$7 x$
							-	$4 x$	-	$4$

Step 3.3.2.1.4.1.5.13

Multiply the new quotient term by the divisor.

				$2 x^{2}$	+	$7 x$	-	$4$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	+	$3 x$
					-	$7 x^{2}$	-	$7 x$
							-	$4 x$	-	$4$
							-	$4 x$	-	$4$

Step 3.3.2.1.4.1.5.14

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

				$2 x^{2}$	+	$7 x$	-	$4$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	+	$3 x$
					-	$7 x^{2}$	-	$7 x$
							-	$4 x$	-	$4$
							+	$4 x$	+	$4$

Step 3.3.2.1.4.1.5.15

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

				$2 x^{2}$	+	$7 x$	-	$4$
$x$	+	$1$		$2 x^{3}$	+	$9 x^{2}$	+	$3 x$	-	$4$
			-	$2 x^{3}$	-	$2 x^{2}$
					+	$7 x^{2}$	+	$3 x$
					-	$7 x^{2}$	-	$7 x$
							-	$4 x$	-	$4$
							+	$4 x$	+	$4$
										$0$

Step 3.3.2.1.4.1.5.16

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

$2 x^{2} + 7 x - 4$

Step 3.3.2.1.4.1.6

Write $2 x^{3} + 9 x^{2} + 3 x - 4$ as a set of factors.

$a = \frac{\frac{(x + 1) (2 x^{2} + 7 x - 4)}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.4.2

Factor by grouping.

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Step 3.3.2.1.4.2.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 4 = - 8$ and whose sum is $b = 7$ .

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Step 3.3.2.1.4.2.1.1

Factor $7$ out of $7 x$ .

$a = \frac{\frac{(x + 1) (2 x^{2} + 7 (x) - 4)}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.4.2.1.2

Rewrite $7$ as $- 1$ plus $8$

$a = \frac{\frac{(x + 1) (2 x^{2} + (- 1 + 8) x - 4)}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.4.2.1.3

Apply the distributive property.

$a = \frac{\frac{(x + 1) (2 x^{2} - 1 x + 8 x - 4)}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.4.2.2

Factor out the greatest common factor from each group.

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Step 3.3.2.1.4.2.2.1

Group the first two terms and the last two terms.

$a = \frac{\frac{(x + 1) ((2 x^{2} - 1 x) + 8 x - 4)}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.4.2.2.2

Factor out the greatest common factor (GCF) from each group.

$a = \frac{\frac{(x + 1) (x (2 x - 1) + 4 (2 x - 1))}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.1.4.2.3

Factor the polynomial by factoring out the greatest common factor, $2 x - 1$ .

$a = \frac{\frac{(x + 1) ((2 x - 1) (x + 4))}{x + 4} - b x - c}{x^{2}}$

$a = \frac{\frac{(x + 1) (2 x - 1) (x + 4)}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.2

Cancel the common factor of $x + 4$ .

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Step 3.3.2.2.1

Cancel the common factor.

$a = \frac{\frac{(x + 1) (2 x - 1) (x + 4)}{x + 4} - b x - c}{x^{2}}$

Step 3.3.2.2.2

Divide $(x + 1) (2 x - 1)$ by $1$ .

$a = \frac{(x + 1) (2 x - 1) - b x - c}{x^{2}}$

Step 3.3.2.3

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

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Step 3.3.2.3.1

Apply the distributive property.

$a = \frac{x (2 x - 1) + 1 (2 x - 1) - b x - c}{x^{2}}$

Step 3.3.2.3.2

Apply the distributive property.

$a = \frac{x (2 x) + x \cdot - 1 + 1 (2 x - 1) - b x - c}{x^{2}}$

Step 3.3.2.3.3

Apply the distributive property.

$a = \frac{x (2 x) + x \cdot - 1 + 1 (2 x) + 1 \cdot - 1 - b x - c}{x^{2}}$

Step 3.3.2.4

Simplify and combine like terms.

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Step 3.3.2.4.1

Simplify each term.

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Step 3.3.2.4.1.1

Rewrite using the commutative property of multiplication.

$a = \frac{2 x \cdot x + x \cdot - 1 + 1 (2 x) + 1 \cdot - 1 - b x - c}{x^{2}}$

Step 3.3.2.4.1.2

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

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Step 3.3.2.4.1.2.1

Move $x$ .

$a = \frac{2 (x \cdot x) + x \cdot - 1 + 1 (2 x) + 1 \cdot - 1 - b x - c}{x^{2}}$

Step 3.3.2.4.1.2.2

Multiply $x$ by $x$ .

$a = \frac{2 x^{2} + x \cdot - 1 + 1 (2 x) + 1 \cdot - 1 - b x - c}{x^{2}}$

Step 3.3.2.4.1.3

Move $- 1$ to the left of $x$ .

$a = \frac{2 x^{2} - 1 \cdot x + 1 (2 x) + 1 \cdot - 1 - b x - c}{x^{2}}$

Step 3.3.2.4.1.4

Rewrite $- 1 x$ as $- x$ .

$a = \frac{2 x^{2} - x + 1 (2 x) + 1 \cdot - 1 - b x - c}{x^{2}}$

Step 3.3.2.4.1.5

Multiply $2 x$ by $1$ .

$a = \frac{2 x^{2} - x + 2 x + 1 \cdot - 1 - b x - c}{x^{2}}$

Step 3.3.2.4.1.6

Multiply $- 1$ by $1$ .

$a = \frac{2 x^{2} - x + 2 x - 1 - b x - c}{x^{2}}$

Step 3.3.2.4.2

Add $- x$ and $2 x$ .

$a = \frac{2 x^{2} + x - 1 - b x - c}{x^{2}}$