Algebra Examples

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

Step 1

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 2 x - 1$ and $b = 4 x + 5$ .

$(2 x - 1 + 4 x + 5) (2 x - 1 - (4 x + 5))$

Step 2

Simplify.

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Step 2.1

Add $2 x$ and $4 x$ .

$(6 x - 1 + 5) (2 x - 1 - (4 x + 5))$

Step 2.2

Add $- 1$ and $5$ .

$(6 x + 4) (2 x - 1 - (4 x + 5))$

Step 2.3

Factor $2$ out of $6 x + 4$ .

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Step 2.3.1

Factor $2$ out of $6 x$ .

$(2 (3 x) + 4) (2 x - 1 - (4 x + 5))$

Step 2.3.2

Factor $2$ out of $4$ .

$(2 (3 x) + 2 (2)) (2 x - 1 - (4 x + 5))$

Step 2.3.3

Factor $2$ out of $2 (3 x) + 2 (2)$ .

$2 (3 x + 2) (2 x - 1 - (4 x + 5))$

Step 2.4

Apply the distributive property.

$2 (3 x + 2) (2 x - 1 - (4 x) - 1 \cdot 5)$

Step 2.5

Multiply $4$ by $- 1$ .

$2 (3 x + 2) (2 x - 1 - 4 x - 1 \cdot 5)$

Step 2.6

Multiply $- 1$ by $5$ .

$2 (3 x + 2) (2 x - 1 - 4 x - 5)$

Step 2.7

Subtract $4 x$ from $2 x$ .

$2 (3 x + 2) (- 2 x - 1 - 5)$

Step 2.8

Subtract $5$ from $- 1$ .

$2 (3 x + 2) (- 2 x - 6)$

Step 2.9

Factor.

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Step 2.9.1

Factor $2$ out of $- 2 x - 6$ .

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Step 2.9.1.1

Factor $2$ out of $- 2 x$ .

$2 (3 x + 2) (2 (- x) - 6)$

Step 2.9.1.2

Factor $2$ out of $- 6$ .

$2 (3 x + 2) (2 (- x) + 2 (- 3))$

Step 2.9.1.3

Factor $2$ out of $2 (- x) + 2 (- 3)$ .

$2 (3 x + 2) (2 (- x - 3))$

Step 2.9.2

Remove unnecessary parentheses.

$2 (3 x + 2) \cdot 2 (- x - 3)$

Step 2.10

Multiply $2$ by $2$ .

$4 (3 x + 2) (- x - 3)$

Step 3

Factor.

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Step 3.1

Factor $- 1$ out of $- x - 3$ .

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Step 3.1.1

Factor $- 1$ out of $- x$ .

$4 (3 x + 2) (- (x) - 3)$

Step 3.1.2

Rewrite $- 3$ as $- 1 (3)$ .

$4 (3 x + 2) (- (x) - 1 (3))$

Step 3.1.3

Factor $- 1$ out of $- (x) - 1 (3)$ .

$4 (3 x + 2) (- (x + 3))$

Step 3.2

Remove unnecessary parentheses.

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

Step 4

Combine exponents.

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Step 4.1

Factor out negative.

$- (4 (3 x + 2) (x + 3))$

Step 4.2

Multiply $4$ by $- 1$ .

$- 4 ((3 x + 2) (x + 3))$

Step 5

Remove unnecessary parentheses.

$- 4 (3 x + 2) (x + 3)$