Solve (a^{2}+4ab+3b^2)^2-(a+b)(a-3b)(a-b)(a+3b)-4ab(2a^2+7ab+6b^2)

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

Evaluate

$$4\left(ab\right)^{2}$$

Solution Steps

Square a^{2}+4ab+3b^{2}.

$$a^{4}+22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-\left(a+b\right)\left(a-3b\right)\left(a-b\right)\left(a+3b\right)-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

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

$$a^{4}+22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-\left(a^{2}-2ab-3b^{2}\right)\left(a-b\right)\left(a+3b\right)-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Use the distributive property to multiply a^{2}-2ab-3b^{2} by a-b and combine like terms.

$$a^{4}+22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-\left(a^{3}-3a^{2}b-ab^{2}+3b^{3}\right)\left(a+3b\right)-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Use the distributive property to multiply a^{3}-3a^{2}b-ab^{2}+3b^{3} by a+3b and combine like terms.

$$a^{4}+22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-\left(a^{4}-10a^{2}b^{2}+9b^{4}\right)-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

To find the opposite of a^{4}-10a^{2}b^{2}+9b^{4}, find the opposite of each term.

$$a^{4}+22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-a^{4}+10a^{2}b^{2}-9b^{4}-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Combine a^{4} and -a^{4} to get 0.

$$22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}+10a^{2}b^{2}-9b^{4}-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Combine 22a^{2}b^{2} and 10a^{2}b^{2} to get 32a^{2}b^{2}.

$$32a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-9b^{4}-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Combine 9b^{4} and -9b^{4} to get 0.

$$32a^{2}b^{2}+24ab^{3}+8ba^{3}-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Use the distributive property to multiply -4ab by 2a^{2}+7ab+6b^{2}.

$$32a^{2}b^{2}+24ab^{3}+8ba^{3}-8a^{3}b-28a^{2}b^{2}-24ab^{3}$$

Combine 8ba^{3} and -8a^{3}b to get 0.

$$32a^{2}b^{2}+24ab^{3}-28a^{2}b^{2}-24ab^{3}$$

Combine 32a^{2}b^{2} and -28a^{2}b^{2} to get 4a^{2}b^{2}.

$$4a^{2}b^{2}+24ab^{3}-24ab^{3}$$

Combine 24ab^{3} and -24ab^{3} to get 0.

$$4a^{2}b^{2}$$

Expand

$$4\left(ab\right)^{2}$$

Solution Steps

Square a^{2}+4ab+3b^{2}.

$$a^{4}+22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-\left(a+b\right)\left(a-3b\right)\left(a-b\right)\left(a+3b\right)-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

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

$$a^{4}+22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-\left(a^{2}-2ab-3b^{2}\right)\left(a-b\right)\left(a+3b\right)-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Use the distributive property to multiply a^{2}-2ab-3b^{2} by a-b and combine like terms.

$$a^{4}+22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-\left(a^{3}-3a^{2}b-ab^{2}+3b^{3}\right)\left(a+3b\right)-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Use the distributive property to multiply a^{3}-3a^{2}b-ab^{2}+3b^{3} by a+3b and combine like terms.

$$a^{4}+22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-\left(a^{4}-10a^{2}b^{2}+9b^{4}\right)-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

To find the opposite of a^{4}-10a^{2}b^{2}+9b^{4}, find the opposite of each term.

$$a^{4}+22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-a^{4}+10a^{2}b^{2}-9b^{4}-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Combine a^{4} and -a^{4} to get 0.

$$22a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}+10a^{2}b^{2}-9b^{4}-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Combine 22a^{2}b^{2} and 10a^{2}b^{2} to get 32a^{2}b^{2}.

$$32a^{2}b^{2}+24ab^{3}+9b^{4}+8ba^{3}-9b^{4}-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Combine 9b^{4} and -9b^{4} to get 0.

$$32a^{2}b^{2}+24ab^{3}+8ba^{3}-4ab\left(2a^{2}+7ab+6b^{2}\right)$$

Use the distributive property to multiply -4ab by 2a^{2}+7ab+6b^{2}.

$$32a^{2}b^{2}+24ab^{3}+8ba^{3}-8a^{3}b-28a^{2}b^{2}-24ab^{3}$$

Combine 8ba^{3} and -8a^{3}b to get 0.

$$32a^{2}b^{2}+24ab^{3}-28a^{2}b^{2}-24ab^{3}$$

Combine 32a^{2}b^{2} and -28a^{2}b^{2} to get 4a^{2}b^{2}.

$$4a^{2}b^{2}+24ab^{3}-24ab^{3}$$

Combine 24ab^{3} and -24ab^{3} to get 0.

$$4a^{2}b^{2}$$