Algebra Examples

$(4 w^{2} - 4 w - 8) - (2 w^{2} + 3 w - 6)$

Step 1

Regroup terms.

$- 4 w + 4 w^{2} - 8 - (2 w^{2} + 3 w - 6)$

Step 2

Apply the distributive property.

$- 4 w + 4 w^{2} - 8 - (2 w^{2}) - (3 w) - - 6$

Step 3

Simplify.

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Step 3.1

Multiply $2$ by $- 1$ .

$- 4 w + 4 w^{2} - 8 - 2 w^{2} - (3 w) - - 6$

Step 3.2

Multiply $3$ by $- 1$ .

$- 4 w + 4 w^{2} - 8 - 2 w^{2} - 3 w - - 6$

Step 3.3

Multiply $- 1$ by $- 6$ .

$- 4 w + 4 w^{2} - 8 - 2 w^{2} - 3 w + 6$

Step 4

Subtract $2 w^{2}$ from $4 w^{2}$ .

$- 4 w + 2 w^{2} - 8 - 3 w + 6$

Step 5

Add $- 8$ and $6$ .

$- 4 w + 2 w^{2} - 3 w - 2$

Step 6

Factor by grouping.

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Step 6.1

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 $a \cdot c = 2 \cdot - 2 = - 4$ and whose sum is $b = - 3$ .

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Step 6.1.1

Factor $- 3$ out of $- 3 w$ .

$- 4 w + 2 w^{2} - 3 (w) - 2$

Step 6.1.2

Rewrite $- 3$ as $1$ plus $- 4$

$- 4 w + 2 w^{2} + (1 - 4) w - 2$

Step 6.1.3

Apply the distributive property.

$- 4 w + 2 w^{2} + 1 w - 4 w - 2$

Step 6.1.4

Multiply $w$ by $1$ .

$- 4 w + 2 w^{2} + w - 4 w - 2$

Step 6.2

Factor out the greatest common factor from each group.

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Step 6.2.1

Group the first two terms and the last two terms.

$- 4 w + (2 w^{2} + w) - 4 w - 2$

Step 6.2.2

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

$- 4 w + w (2 w + 1) - 2 (2 w + 1)$

Step 6.3

Factor the polynomial by factoring out the greatest common factor, $2 w + 1$ .

$- 4 w + (2 w + 1) (w - 2)$

Step 7

Factor $- 1$ out of $- 4 w + (2 w + 1) (w - 2)$ .

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Step 7.1

Reorder $- 4 w$ and $(2 w + 1) (w - 2)$ .

$(2 w + 1) (w - 2) - 4 w$

Step 7.2

Factor $- 1$ out of $(2 w + 1) (w - 2)$ .

$- ((- 2 w - 1) (w - 2)) - 4 w$

Step 7.3

Factor $- 1$ out of $- 4 w$ .

$- ((- 2 w - 1) (w - 2)) - (4 w)$

Step 7.4

Factor $- 1$ out of $- ((- 2 w - 1) (w - 2)) - (4 w)$ .

$- ((- 2 w - 1) (w - 2) + 4 w)$

Step 8

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

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Step 8.1

Apply the distributive property.

$- (- 2 w (w - 2) - 1 (w - 2) + 4 w)$

Step 8.2

Apply the distributive property.

$- (- 2 w \cdot w - 2 w \cdot - 2 - 1 (w - 2) + 4 w)$

Step 8.3

Apply the distributive property.

$- (- 2 w \cdot w - 2 w \cdot - 2 - 1 w - 1 \cdot - 2 + 4 w)$

Step 9

Simplify and combine like terms.

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Step 9.1

Simplify each term.

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Step 9.1.1

Multiply $w$ by $w$ by adding the exponents.

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Step 9.1.1.1

Move $w$ .

$- (- 2 (w \cdot w) - 2 w \cdot - 2 - 1 w - 1 \cdot - 2 + 4 w)$

Step 9.1.1.2

Multiply $w$ by $w$ .

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

Step 9.1.2

Multiply $- 2$ by $- 2$ .

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

Step 9.1.3

Rewrite $- 1 w$ as $- w$ .

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

Step 9.1.4

Multiply $- 1$ by $- 2$ .

$- (- 2 w^{2} + 4 w - w + 2 + 4 w)$

Step 9.2

Subtract $w$ from $4 w$ .

$- (- 2 w^{2} + 3 w + 2 + 4 w)$

Step 10

Add $3 w$ and $4 w$ .

$- (- 2 w^{2} + 7 w + 2)$

Step 11

Factor.

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Step 11.1

Factor $- 1$ out of $- 2 w^{2} + 7 w + 2$ .

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Step 11.1.1

Factor $- 1$ out of $- 2 w^{2}$ .

$- (- (2 w^{2}) + 7 w + 2)$

Step 11.1.2

Factor $- 1$ out of $7 w$ .

$- (- (2 w^{2}) - (- 7 w) + 2)$

Step 11.1.3

Rewrite $2$ as $- 1 (- 2)$ .

$- (- (2 w^{2}) - (- 7 w) - 1 (- 2))$

Step 11.1.4

Factor $- 1$ out of $- (2 w^{2}) - (- 7 w)$ .

$- (- (2 w^{2} - 7 w) - 1 (- 2))$

Step 11.1.5

Factor $- 1$ out of $- (2 w^{2} - 7 w) - 1 (- 2)$ .

$- - (2 w^{2} - 7 w - 2)$

Step 11.2

Remove unnecessary parentheses.

$- - 1 (2 w^{2} - 7 w - 2)$

Step 12

Combine exponents.

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Step 12.1

Factor out negative.

$- - (2 w^{2} - 7 w - 2)$

Step 12.2

Multiply $- 1$ by $- 1$ .

$1 (2 w^{2} - 7 w - 2)$

Step 12.3

Multiply $2 w^{2} - 7 w - 2$ by $1$ .

$2 w^{2} - 7 w - 2$