Solve (2a+3b)^2-2(2a+3b)(5b-4a)+(4a-5b)^2

$$( 2 a + 3 b ) ^ { 2 } - 2 ( 2 a + 3 b ) ( 5 b - 4 a ) + ( 4 a - 5 b ) ^ { 2 }$$

Evaluate

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

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(2a+3b\right)^{2}.

$$4a^{2}+12ab+9b^{2}-2\left(2a+3b\right)\left(5b-4a\right)+\left(4a-5b\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(4a-5b\right)^{2}.

$$4a^{2}+12ab+9b^{2}-2\left(2a+3b\right)\left(5b-4a\right)+16a^{2}-40ab+25b^{2}$$

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

$$4a^{2}+12ab+9b^{2}+\left(-4a-6b\right)\left(5b-4a\right)+16a^{2}-40ab+25b^{2}$$

Use the distributive property to multiply -4a-6b by 5b-4a and combine like terms.

$$4a^{2}+12ab+9b^{2}+4ab+16a^{2}-30b^{2}+16a^{2}-40ab+25b^{2}$$

Combine 12ab and 4ab to get 16ab.

$$4a^{2}+16ab+9b^{2}+16a^{2}-30b^{2}+16a^{2}-40ab+25b^{2}$$

Combine 4a^{2} and 16a^{2} to get 20a^{2}.

$$20a^{2}+16ab+9b^{2}-30b^{2}+16a^{2}-40ab+25b^{2}$$

Combine 9b^{2} and -30b^{2} to get -21b^{2}.

$$20a^{2}+16ab-21b^{2}+16a^{2}-40ab+25b^{2}$$

Combine 20a^{2} and 16a^{2} to get 36a^{2}.

$$36a^{2}+16ab-21b^{2}-40ab+25b^{2}$$

Combine 16ab and -40ab to get -24ab.

$$36a^{2}-24ab-21b^{2}+25b^{2}$$

Combine -21b^{2} and 25b^{2} to get 4b^{2}.

$$36a^{2}-24ab+4b^{2}$$

Expand

$$36a^{2}-24ab+4b^{2}$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(2a+3b\right)^{2}.

$$4a^{2}+12ab+9b^{2}-2\left(2a+3b\right)\left(5b-4a\right)+\left(4a-5b\right)^{2}$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(4a-5b\right)^{2}.

$$4a^{2}+12ab+9b^{2}-2\left(2a+3b\right)\left(5b-4a\right)+16a^{2}-40ab+25b^{2}$$

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

$$4a^{2}+12ab+9b^{2}+\left(-4a-6b\right)\left(5b-4a\right)+16a^{2}-40ab+25b^{2}$$

Use the distributive property to multiply -4a-6b by 5b-4a and combine like terms.

$$4a^{2}+12ab+9b^{2}+4ab+16a^{2}-30b^{2}+16a^{2}-40ab+25b^{2}$$

Combine 12ab and 4ab to get 16ab.

$$4a^{2}+16ab+9b^{2}+16a^{2}-30b^{2}+16a^{2}-40ab+25b^{2}$$

Combine 4a^{2} and 16a^{2} to get 20a^{2}.

$$20a^{2}+16ab+9b^{2}-30b^{2}+16a^{2}-40ab+25b^{2}$$

Combine 9b^{2} and -30b^{2} to get -21b^{2}.

$$20a^{2}+16ab-21b^{2}+16a^{2}-40ab+25b^{2}$$

Combine 20a^{2} and 16a^{2} to get 36a^{2}.

$$36a^{2}+16ab-21b^{2}-40ab+25b^{2}$$

Combine 16ab and -40ab to get -24ab.

$$36a^{2}-24ab-21b^{2}+25b^{2}$$

Combine -21b^{2} and 25b^{2} to get 4b^{2}.

$$36a^{2}-24ab+4b^{2}$$