Solve (1-2b)^2+2(b-3)(2+3b)-(b-2)^2=

$$( 1 - 2 b ) ^ { 2 } + 2 ( b - 3 ) ( 2 + 3 b ) - ( b - 2 ) ^ { 2 } =$$

Evaluate

$$9b^{2}-14b-15$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(1-2b\right)^{2}.

$$1-4b+4b^{2}+2\left(b-3\right)\left(2+3b\right)-\left(b-2\right)^{2}$$

Use the distributive property to multiply 2 by b-3.

$$1-4b+4b^{2}+\left(2b-6\right)\left(2+3b\right)-\left(b-2\right)^{2}$$

Use the distributive property to multiply 2b-6 by 2+3b and combine like terms.

$$1-4b+4b^{2}-14b+6b^{2}-12-\left(b-2\right)^{2}$$

Combine -4b and -14b to get -18b.

$$1-18b+4b^{2}+6b^{2}-12-\left(b-2\right)^{2}$$

Combine 4b^{2} and 6b^{2} to get 10b^{2}.

$$1-18b+10b^{2}-12-\left(b-2\right)^{2}$$

Subtract 12 from 1 to get -11.

$$-11-18b+10b^{2}-\left(b-2\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(b-2\right)^{2}.

$$-11-18b+10b^{2}-\left(b^{2}-4b+4\right)$$

To find the opposite of b^{2}-4b+4, find the opposite of each term.

$$-11-18b+10b^{2}-b^{2}+4b-4$$

Combine 10b^{2} and -b^{2} to get 9b^{2}.

$$-11-18b+9b^{2}+4b-4$$

Combine -18b and 4b to get -14b.

$$-11-14b+9b^{2}-4$$

Subtract 4 from -11 to get -15.

$$-15-14b+9b^{2}$$

Expand

$$9b^{2}-14b-15$$

Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(1-2b\right)^{2}.

$$1-4b+4b^{2}+2\left(b-3\right)\left(2+3b\right)-\left(b-2\right)^{2}$$

Use the distributive property to multiply 2 by b-3.

$$1-4b+4b^{2}+\left(2b-6\right)\left(2+3b\right)-\left(b-2\right)^{2}$$

Use the distributive property to multiply 2b-6 by 2+3b and combine like terms.

$$1-4b+4b^{2}-14b+6b^{2}-12-\left(b-2\right)^{2}$$

Combine -4b and -14b to get -18b.

$$1-18b+4b^{2}+6b^{2}-12-\left(b-2\right)^{2}$$

Combine 4b^{2} and 6b^{2} to get 10b^{2}.

$$1-18b+10b^{2}-12-\left(b-2\right)^{2}$$

Subtract 12 from 1 to get -11.

$$-11-18b+10b^{2}-\left(b-2\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(b-2\right)^{2}.

$$-11-18b+10b^{2}-\left(b^{2}-4b+4\right)$$

To find the opposite of b^{2}-4b+4, find the opposite of each term.

$$-11-18b+10b^{2}-b^{2}+4b-4$$

Combine 10b^{2} and -b^{2} to get 9b^{2}.

$$-11-18b+9b^{2}+4b-4$$

Combine -18b and 4b to get -14b.

$$-11-14b+9b^{2}-4$$

Subtract 4 from -11 to get -15.

$$-15-14b+9b^{2}$$