Solve 2left(a+bright)^2+left(a+bright)left(a-bright)-6left(a-bright)^2

$$2 { \left(a+b \right) }^{ 2 } + \left( a+b \right) \left( a-b \right) -6 { \left( a-b \right) }^{ 2 }$$

Evaluate

$$\left(3a-b\right)\left(5b-a\right)$$

Solution Steps

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

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

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

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

Consider \left(a+b\right)\left(a-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}.

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

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

$$3a^{2}+4ab+2b^{2}-b^{2}-6\left(a-b\right)^{2}$$

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

$$3a^{2}+4ab+b^{2}-6\left(a-b\right)^{2}$$

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

$$3a^{2}+4ab+b^{2}-6\left(a^{2}-2ab+b^{2}\right)$$

Use the distributive property to multiply -6 by a^{2}-2ab+b^{2}.

$$3a^{2}+4ab+b^{2}-6a^{2}+12ab-6b^{2}$$

Combine 3a^{2} and -6a^{2} to get -3a^{2}.

$$-3a^{2}+4ab+b^{2}+12ab-6b^{2}$$

Combine 4ab and 12ab to get 16ab.

$$-3a^{2}+16ab+b^{2}-6b^{2}$$

Combine b^{2} and -6b^{2} to get -5b^{2}.

$$-3a^{2}+16ab-5b^{2}$$

Expand

$$-3a^{2}+16ab-5b^{2}$$

Solution Steps

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

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

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

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

Consider \left(a+b\right)\left(a-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}.

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

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

$$3a^{2}+4ab+2b^{2}-b^{2}-6\left(a-b\right)^{2}$$

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

$$3a^{2}+4ab+b^{2}-6\left(a-b\right)^{2}$$

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

$$3a^{2}+4ab+b^{2}-6\left(a^{2}-2ab+b^{2}\right)$$

Use the distributive property to multiply -6 by a^{2}-2ab+b^{2}.

$$3a^{2}+4ab+b^{2}-6a^{2}+12ab-6b^{2}$$

Combine 3a^{2} and -6a^{2} to get -3a^{2}.

$$-3a^{2}+4ab+b^{2}+12ab-6b^{2}$$

Combine 4ab and 12ab to get 16ab.

$$-3a^{2}+16ab+b^{2}-6b^{2}$$

Combine b^{2} and -6b^{2} to get -5b^{2}.

$$-3a^{2}+16ab-5b^{2}$$