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

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

Evaluate

$$2b\left(2a+3b\right)$$

Solution Steps

Consider \left(4a-5b\right)\left(4a+5b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(4a\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$16a^{2}-\left(5b\right)^{2}-\left(4a+2b\right)\left(4a-3b\right)+\left(-5b\right)^{2}$$

Expand \left(5b\right)^{2}.

$$16a^{2}-5^{2}b^{2}-\left(4a+2b\right)\left(4a-3b\right)+\left(-5b\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

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

Use the distributive property to multiply 4a+2b by 4a-3b and combine like terms.

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

To find the opposite of 16a^{2}-4ab-6b^{2}, find the opposite of each term.

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

Combine 16a^{2} and -16a^{2} to get 0.

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

Combine -25b^{2} and 6b^{2} to get -19b^{2}.

$$-19b^{2}+4ab+\left(-5b\right)^{2}$$

Expand \left(-5b\right)^{2}.

$$-19b^{2}+4ab+\left(-5\right)^{2}b^{2}$$

Calculate -5 to the power of 2 and get 25.

$$-19b^{2}+4ab+25b^{2}$$

Combine -19b^{2} and 25b^{2} to get 6b^{2}.

$$6b^{2}+4ab$$

Expand

$$4ab+6b^{2}$$

Solution Steps

Consider \left(4a-5b\right)\left(4a+5b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

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

Expand \left(4a\right)^{2}.

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

Calculate 4 to the power of 2 and get 16.

$$16a^{2}-\left(5b\right)^{2}-\left(4a+2b\right)\left(4a-3b\right)+\left(-5b\right)^{2}$$

Expand \left(5b\right)^{2}.

$$16a^{2}-5^{2}b^{2}-\left(4a+2b\right)\left(4a-3b\right)+\left(-5b\right)^{2}$$

Calculate 5 to the power of 2 and get 25.

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

Use the distributive property to multiply 4a+2b by 4a-3b and combine like terms.

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

To find the opposite of 16a^{2}-4ab-6b^{2}, find the opposite of each term.

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

Combine 16a^{2} and -16a^{2} to get 0.

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

Combine -25b^{2} and 6b^{2} to get -19b^{2}.

$$-19b^{2}+4ab+\left(-5b\right)^{2}$$

Expand \left(-5b\right)^{2}.

$$-19b^{2}+4ab+\left(-5\right)^{2}b^{2}$$

Calculate -5 to the power of 2 and get 25.

$$-19b^{2}+4ab+25b^{2}$$

Combine -19b^{2} and 25b^{2} to get 6b^{2}.

$$6b^{2}+4ab$$