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

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

Evaluate

$$4b\left(3a+2b\right)$$

Solution Steps

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

$$9a^{2}+12ab+4b^{2}-\left(3a+2b\right)\left(3a-2b\right)$$

Consider \left(3a+2b\right)\left(3a-2b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$9a^{2}+12ab+4b^{2}-\left(\left(3a\right)^{2}-\left(2b\right)^{2}\right)$$

Expand \left(3a\right)^{2}.

$$9a^{2}+12ab+4b^{2}-\left(3^{2}a^{2}-\left(2b\right)^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$9a^{2}+12ab+4b^{2}-\left(9a^{2}-\left(2b\right)^{2}\right)$$

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

$$9a^{2}+12ab+4b^{2}-\left(9a^{2}-2^{2}b^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$9a^{2}+12ab+4b^{2}-\left(9a^{2}-4b^{2}\right)$$

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

$$9a^{2}+12ab+4b^{2}-9a^{2}+4b^{2}$$

Combine 9a^{2} and -9a^{2} to get 0.

$$12ab+4b^{2}+4b^{2}$$

Combine 4b^{2} and 4b^{2} to get 8b^{2}.

$$12ab+8b^{2}$$

Expand

$$12ab+8b^{2}$$

Solution Steps

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

$$9a^{2}+12ab+4b^{2}-\left(3a+2b\right)\left(3a-2b\right)$$

Consider \left(3a+2b\right)\left(3a-2b\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$9a^{2}+12ab+4b^{2}-\left(\left(3a\right)^{2}-\left(2b\right)^{2}\right)$$

Expand \left(3a\right)^{2}.

$$9a^{2}+12ab+4b^{2}-\left(3^{2}a^{2}-\left(2b\right)^{2}\right)$$

Calculate 3 to the power of 2 and get 9.

$$9a^{2}+12ab+4b^{2}-\left(9a^{2}-\left(2b\right)^{2}\right)$$

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

$$9a^{2}+12ab+4b^{2}-\left(9a^{2}-2^{2}b^{2}\right)$$

Calculate 2 to the power of 2 and get 4.

$$9a^{2}+12ab+4b^{2}-\left(9a^{2}-4b^{2}\right)$$

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

$$9a^{2}+12ab+4b^{2}-9a^{2}+4b^{2}$$

Combine 9a^{2} and -9a^{2} to get 0.

$$12ab+4b^{2}+4b^{2}$$

Combine 4b^{2} and 4b^{2} to get 8b^{2}.

$$12ab+8b^{2}$$