Solve (3a+6)^2-(a+b)*(a-2b)+(a-b)^2=

$$( 3 a + 6 ) ^ { 2 } - ( a + b ) \cdot ( a - 2 b ) + ( a - b ) ^ { 2 } =$$

Evaluate

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

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(3a+6\right)^{2}.

$$9a^{2}+36a+36-\left(a+b\right)\left(a-2b\right)+\left(a-b\right)^{2}$$

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

$$9a^{2}+36a+36-\left(a^{2}-ab-2b^{2}\right)+\left(a-b\right)^{2}$$

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

$$9a^{2}+36a+36-a^{2}+ab+2b^{2}+\left(a-b\right)^{2}$$

Combine 9a^{2} and -a^{2} to get 8a^{2}.

$$8a^{2}+36a+36+ab+2b^{2}+\left(a-b\right)^{2}$$

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

$$8a^{2}+36a+36+ab+2b^{2}+a^{2}-2ab+b^{2}$$

Combine 8a^{2} and a^{2} to get 9a^{2}.

$$9a^{2}+36a+36+ab+2b^{2}-2ab+b^{2}$$

Combine ab and -2ab to get -ab.

$$9a^{2}+36a+36-ab+2b^{2}+b^{2}$$

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

$$9a^{2}+36a+36-ab+3b^{2}$$

Expand

$$9a^{2}-ab+36a+3b^{2}+36$$

Solution Steps

Use binomial theorem \left(p+q\right)^{2}=p^{2}+2pq+q^{2} to expand \left(3a+6\right)^{2}.

$$9a^{2}+36a+36-\left(a+b\right)\left(a-2b\right)+\left(a-b\right)^{2}$$

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

$$9a^{2}+36a+36-\left(a^{2}-ab-2b^{2}\right)+\left(a-b\right)^{2}$$

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

$$9a^{2}+36a+36-a^{2}+ab+2b^{2}+\left(a-b\right)^{2}$$

Combine 9a^{2} and -a^{2} to get 8a^{2}.

$$8a^{2}+36a+36+ab+2b^{2}+\left(a-b\right)^{2}$$

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

$$8a^{2}+36a+36+ab+2b^{2}+a^{2}-2ab+b^{2}$$

Combine 8a^{2} and a^{2} to get 9a^{2}.

$$9a^{2}+36a+36+ab+2b^{2}-2ab+b^{2}$$

Combine ab and -2ab to get -ab.

$$9a^{2}+36a+36-ab+2b^{2}+b^{2}$$

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

$$9a^{2}+36a+36-ab+3b^{2}$$