Factor 2(3-b)+5(b-3)^2

$2 (3 - b) + 5 {(b - 3)}^{2}$

Step 1

Apply the distributive property.

$2 \cdot 3 + 2 (- b) + 5 {(b - 3)}^{2}$

Step 2

Multiply $2$ by $3$ .

$6 + 2 (- b) + 5 {(b - 3)}^{2}$

Step 3

Multiply $- 1$ by $2$ .

$6 - 2 b + 5 {(b - 3)}^{2}$

Step 4

Rewrite ${(b - 3)}^{2}$ as $(b - 3) (b - 3)$ .

$6 - 2 b + 5 ((b - 3) (b - 3))$

Step 5

Expand $(b - 3) (b - 3)$ using the FOIL Method.

Apply the distributive property.

$6 - 2 b + 5 (b (b - 3) - 3 (b - 3))$

Apply the distributive property.

$6 - 2 b + 5 (b \cdot b + b \cdot - 3 - 3 (b - 3))$

Apply the distributive property.

$6 - 2 b + 5 (b \cdot b + b \cdot - 3 - 3 b - 3 \cdot - 3)$

Step 6

Simplify and combine like terms.

Simplify each term.

Multiply $b$ by $b$ .

$6 - 2 b + 5 (b^{2} + b \cdot - 3 - 3 b - 3 \cdot - 3)$

Move $- 3$ to the left of $b$ .

$6 - 2 b + 5 (b^{2} - 3 b - 3 b - 3 \cdot - 3)$

Multiply $- 3$ by $- 3$ .

$6 - 2 b + 5 (b^{2} - 3 b - 3 b + 9)$

Subtract $3 b$ from $- 3 b$ .

$6 - 2 b + 5 (b^{2} - 6 b + 9)$

Step 7

Apply the distributive property.

$6 - 2 b + 5 b^{2} + 5 (- 6 b) + 5 \cdot 9$

Step 8

Simplify.

Multiply $- 6$ by $5$ .

$6 - 2 b + 5 b^{2} - 30 b + 5 \cdot 9$

Multiply $5$ by $9$ .

$6 - 2 b + 5 b^{2} - 30 b + 45$

Step 9

Add $6$ and $45$ .

$- 2 b + 5 b^{2} - 30 b + 51$

Step 10

Subtract $30 b$ from $- 2 b$ .

$5 b^{2} - 32 b + 51$

Step 11

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 $a \cdot c = 5 \cdot 51 = 255$ and whose sum is $b = - 32$ .

Factor $- 32$ out of $- 32 b$ .

$5 b^{2} - 32 (b) + 51$

Rewrite $- 32$ as $- 15$ plus $- 17$

$5 b^{2} + (- 15 - 17) b + 51$

Apply the distributive property.

$5 b^{2} - 15 b - 17 b + 51$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$(5 b^{2} - 15 b) - 17 b + 51$

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

$5 b (b - 3) - 17 (b - 3)$

Factor the polynomial by factoring out the greatest common factor, $b - 3$ .

$(b - 3) (5 b - 17)$

Factor 2(3-b)+5(b-3)^2

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