Factor by Grouping -3b(b-2)+7(b-2)^2

$- 3 b (b - 2) + 7 {(b - 2)}^{2}$

Step 1

Simplify each term.

Apply the distributive property.

$- 3 b \cdot b - 3 b \cdot - 2 + 7 {(b - 2)}^{2}$

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

$- 3 (b \cdot b) - 3 b \cdot - 2 + 7 {(b - 2)}^{2}$

Multiply $b$ by $b$ .

$- 3 b^{2} - 3 b \cdot - 2 + 7 {(b - 2)}^{2}$

Multiply $- 2$ by $- 3$ .

$- 3 b^{2} + 6 b + 7 {(b - 2)}^{2}$

Rewrite ${(b - 2)}^{2}$ as $(b - 2) (b - 2)$ .

$- 3 b^{2} + 6 b + 7 ((b - 2) (b - 2))$

Expand $(b - 2) (b - 2)$ using the FOIL Method.

Apply the distributive property.

$- 3 b^{2} + 6 b + 7 (b (b - 2) - 2 (b - 2))$

Apply the distributive property.

$- 3 b^{2} + 6 b + 7 (b \cdot b + b \cdot - 2 - 2 (b - 2))$

Apply the distributive property.

$- 3 b^{2} + 6 b + 7 (b \cdot b + b \cdot - 2 - 2 b - 2 \cdot - 2)$

Simplify and combine like terms.

Simplify each term.

Multiply $b$ by $b$ .

$- 3 b^{2} + 6 b + 7 (b^{2} + b \cdot - 2 - 2 b - 2 \cdot - 2)$

Move $- 2$ to the left of $b$ .

$- 3 b^{2} + 6 b + 7 (b^{2} - 2 \cdot b - 2 b - 2 \cdot - 2)$

Multiply $- 2$ by $- 2$ .

$- 3 b^{2} + 6 b + 7 (b^{2} - 2 b - 2 b + 4)$

Subtract $2 b$ from $- 2 b$ .

$- 3 b^{2} + 6 b + 7 (b^{2} - 4 b + 4)$

Apply the distributive property.

$- 3 b^{2} + 6 b + 7 b^{2} + 7 (- 4 b) + 7 \cdot 4$

Simplify.

Multiply $- 4$ by $7$ .

$- 3 b^{2} + 6 b + 7 b^{2} - 28 b + 7 \cdot 4$

Multiply $7$ by $4$ .

$- 3 b^{2} + 6 b + 7 b^{2} - 28 b + 28$

Step 2

Simplify by adding terms.

Add $- 3 b^{2}$ and $7 b^{2}$ .

$4 b^{2} + 6 b - 28 b + 28$

Subtract $28 b$ from $6 b$ .

$4 b^{2} - 22 b + 28$

Step 3

Factor out the GCF of $2$ from $4 b^{2} - 22 b + 28$ .

Factor out the GCF of $2$ from each term in the polynomial.

Factor out the GCF of $2$ from the expression $4 b^{2}$ .

$2 (2 b^{2}) - 22 b + 28$

Factor out the GCF of $2$ from the expression $- 22 b$ .

$2 (2 b^{2}) + 2 (- 11 b) + 28$

Factor out the GCF of $2$ from the expression $28$ .

$2 (2 b^{2}) + 2 (- 11 b) + 2 (14)$

Since all the terms share a common factor of $2$ , it can be factored out of each term.

$2 (2 b^{2} - 11 b + 14)$

Step 4

Factor by grouping.

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 $a \cdot c = 2 \cdot 14 = 28$ and whose sum is $b = - 11$ .

Factor $- 11$ out of $- 11 b$ .

$2 (2 b^{2} - 11 b + 14)$

Rewrite $- 11$ as $- 4$ plus $- 7$

$2 (2 b^{2} + (- 4 - 7) b + 14)$

Apply the distributive property.

$2 (2 b^{2} - 4 b - 7 b + 14)$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$2 ((2 b^{2} - 4 b) - 7 b + 14)$

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

$2 (2 b (b - 2) - 7 (b - 2))$

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

$2 ((b - 2) (2 b - 7))$

Factor by Grouping -3b(b-2)+7(b-2)^2

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