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

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

Evaluate

$$8ab+17$$

Solution Steps

Use the distributive property to multiply 5-ab by 5+ab and combine like terms.

$$25-a^{2}b^{2}-2\left(ab-2\right)^{2}+3a^{2}b^{2}$$

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

$$25-a^{2}b^{2}-2\left(a^{2}b^{2}-4ab+4\right)+3a^{2}b^{2}$$

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

$$25-a^{2}b^{2}-2a^{2}b^{2}+8ab-8+3a^{2}b^{2}$$

Combine -a^{2}b^{2} and -2a^{2}b^{2} to get -3a^{2}b^{2}.

$$25-3a^{2}b^{2}+8ab-8+3a^{2}b^{2}$$

Subtract 8 from 25 to get 17.

$$17-3a^{2}b^{2}+8ab+3a^{2}b^{2}$$

Combine -3a^{2}b^{2} and 3a^{2}b^{2} to get 0.

$$17+8ab$$

Expand

$$8ab+17$$

Solution Steps

Use the distributive property to multiply 5-ab by 5+ab and combine like terms.

$$25-a^{2}b^{2}-2\left(ab-2\right)^{2}+3a^{2}b^{2}$$

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

$$25-a^{2}b^{2}-2\left(a^{2}b^{2}-4ab+4\right)+3a^{2}b^{2}$$

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

$$25-a^{2}b^{2}-2a^{2}b^{2}+8ab-8+3a^{2}b^{2}$$

Combine -a^{2}b^{2} and -2a^{2}b^{2} to get -3a^{2}b^{2}.

$$25-3a^{2}b^{2}+8ab-8+3a^{2}b^{2}$$

Subtract 8 from 25 to get 17.

$$17-3a^{2}b^{2}+8ab+3a^{2}b^{2}$$

Combine -3a^{2}b^{2} and 3a^{2}b^{2} to get 0.

$$17+8ab$$