Solve (a+b)(a-b)[(2a-3b+5)^{2}-(3a-2b)^2-5(4a-6b+5)]-5b^2(2a^2-b^2)

$$( a + b ) ( a - b ) [ ( 2 a - 3 b + 5 ) ^ { 2 } - ( 3 a - 2 b ) ^ { 2 } - 5 ( 4 a - 6 b + 5 ) ] - 5 b ^ { 2 } ( 2 a ^ { 2 } - b ^ { 2 } )$$

Evaluate

$$-5a^{4}$$

Solution Steps

Square 2a-3b+5.

$$\left(a+b\right)\left(a-b\right)\left(4a^{2}-12ab+20a+9b^{2}-30b+25-\left(3a-2b\right)^{2}-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

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

$$\left(a+b\right)\left(a-b\right)\left(4a^{2}-12ab+20a+9b^{2}-30b+25-\left(9a^{2}-12ab+4b^{2}\right)-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

To find the opposite of 9a^{2}-12ab+4b^{2}, find the opposite of each term.

$$\left(a+b\right)\left(a-b\right)\left(4a^{2}-12ab+20a+9b^{2}-30b+25-9a^{2}+12ab-4b^{2}-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Combine 4a^{2} and -9a^{2} to get -5a^{2}.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}-12ab+20a+9b^{2}-30b+25+12ab-4b^{2}-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Combine -12ab and 12ab to get 0.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+20a+9b^{2}-30b+25-4b^{2}-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Combine 9b^{2} and -4b^{2} to get 5b^{2}.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+20a+5b^{2}-30b+25-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Use the distributive property to multiply -5 by 4a-6b+5.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+20a+5b^{2}-30b+25-20a+30b-25\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Combine 20a and -20a to get 0.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+5b^{2}-30b+25+30b-25\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Combine -30b and 30b to get 0.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+5b^{2}+25-25\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Subtract 25 from 25 to get 0.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+5b^{2}\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Use the distributive property to multiply a+b by a-b and combine like terms.

$$\left(a^{2}-b^{2}\right)\left(-5a^{2}+5b^{2}\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Use the distributive property to multiply a^{2}-b^{2} by -5a^{2}+5b^{2} and combine like terms.

$$-5a^{4}+10a^{2}b^{2}-5b^{4}-5b^{2}\left(2a^{2}-b^{2}\right)$$

Use the distributive property to multiply -5b^{2} by 2a^{2}-b^{2}.

$$-5a^{4}+10a^{2}b^{2}-5b^{4}-10b^{2}a^{2}+5b^{4}$$

Combine 10a^{2}b^{2} and -10b^{2}a^{2} to get 0.

$$-5a^{4}-5b^{4}+5b^{4}$$

Combine -5b^{4} and 5b^{4} to get 0.

$$-5a^{4}$$

Expand

$$-5a^{4}$$

Solution Steps

Square 2a-3b+5.

$$\left(a+b\right)\left(a-b\right)\left(4a^{2}-12ab+20a+9b^{2}-30b+25-\left(3a-2b\right)^{2}-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

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

$$\left(a+b\right)\left(a-b\right)\left(4a^{2}-12ab+20a+9b^{2}-30b+25-\left(9a^{2}-12ab+4b^{2}\right)-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

To find the opposite of 9a^{2}-12ab+4b^{2}, find the opposite of each term.

$$\left(a+b\right)\left(a-b\right)\left(4a^{2}-12ab+20a+9b^{2}-30b+25-9a^{2}+12ab-4b^{2}-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Combine 4a^{2} and -9a^{2} to get -5a^{2}.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}-12ab+20a+9b^{2}-30b+25+12ab-4b^{2}-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Combine -12ab and 12ab to get 0.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+20a+9b^{2}-30b+25-4b^{2}-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Combine 9b^{2} and -4b^{2} to get 5b^{2}.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+20a+5b^{2}-30b+25-5\left(4a-6b+5\right)\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Use the distributive property to multiply -5 by 4a-6b+5.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+20a+5b^{2}-30b+25-20a+30b-25\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Combine 20a and -20a to get 0.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+5b^{2}-30b+25+30b-25\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Combine -30b and 30b to get 0.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+5b^{2}+25-25\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Subtract 25 from 25 to get 0.

$$\left(a+b\right)\left(a-b\right)\left(-5a^{2}+5b^{2}\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Use the distributive property to multiply a+b by a-b and combine like terms.

$$\left(a^{2}-b^{2}\right)\left(-5a^{2}+5b^{2}\right)-5b^{2}\left(2a^{2}-b^{2}\right)$$

Use the distributive property to multiply a^{2}-b^{2} by -5a^{2}+5b^{2} and combine like terms.

$$-5a^{4}+10a^{2}b^{2}-5b^{4}-5b^{2}\left(2a^{2}-b^{2}\right)$$

Use the distributive property to multiply -5b^{2} by 2a^{2}-b^{2}.

$$-5a^{4}+10a^{2}b^{2}-5b^{4}-10b^{2}a^{2}+5b^{4}$$

Combine 10a^{2}b^{2} and -10b^{2}a^{2} to get 0.

$$-5a^{4}-5b^{4}+5b^{4}$$

Combine -5b^{4} and 5b^{4} to get 0.

$$-5a^{4}$$