Solve (2a+b)^2-(5a+b)(5a-b)+3a(7a-b)

$$( 2 a + b ) ^ { 2 } - ( 5 a + b ) ( 5 a - b ) + 3 a ( 7 a - b )$$

Evaluate

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

Solution Steps

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

$$4a^{2}+4ab+b^{2}-\left(5a+b\right)\left(5a-b\right)+3a\left(7a-b\right)$$

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

$$4a^{2}+4ab+b^{2}-\left(\left(5a\right)^{2}-b^{2}\right)+3a\left(7a-b\right)$$

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

$$4a^{2}+4ab+b^{2}-\left(5^{2}a^{2}-b^{2}\right)+3a\left(7a-b\right)$$

Calculate 5 to the power of 2 and get 25.

$$4a^{2}+4ab+b^{2}-\left(25a^{2}-b^{2}\right)+3a\left(7a-b\right)$$

To find the opposite of 25a^{2}-b^{2}, find the opposite of each term.

$$4a^{2}+4ab+b^{2}-25a^{2}+b^{2}+3a\left(7a-b\right)$$

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

$$-21a^{2}+4ab+b^{2}+b^{2}+3a\left(7a-b\right)$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$-21a^{2}+4ab+2b^{2}+3a\left(7a-b\right)$$

Use the distributive property to multiply 3a by 7a-b.

$$-21a^{2}+4ab+2b^{2}+21a^{2}-3ab$$

Combine -21a^{2} and 21a^{2} to get 0.

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

Combine 4ab and -3ab to get ab.

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

Expand

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

Solution Steps

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

$$4a^{2}+4ab+b^{2}-\left(5a+b\right)\left(5a-b\right)+3a\left(7a-b\right)$$

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

$$4a^{2}+4ab+b^{2}-\left(\left(5a\right)^{2}-b^{2}\right)+3a\left(7a-b\right)$$

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

$$4a^{2}+4ab+b^{2}-\left(5^{2}a^{2}-b^{2}\right)+3a\left(7a-b\right)$$

Calculate 5 to the power of 2 and get 25.

$$4a^{2}+4ab+b^{2}-\left(25a^{2}-b^{2}\right)+3a\left(7a-b\right)$$

To find the opposite of 25a^{2}-b^{2}, find the opposite of each term.

$$4a^{2}+4ab+b^{2}-25a^{2}+b^{2}+3a\left(7a-b\right)$$

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

$$-21a^{2}+4ab+b^{2}+b^{2}+3a\left(7a-b\right)$$

Combine b^{2} and b^{2} to get 2b^{2}.

$$-21a^{2}+4ab+2b^{2}+3a\left(7a-b\right)$$

Use the distributive property to multiply 3a by 7a-b.

$$-21a^{2}+4ab+2b^{2}+21a^{2}-3ab$$

Combine -21a^{2} and 21a^{2} to get 0.

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

Combine 4ab and -3ab to get ab.

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