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

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

Evaluate

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

Solution Steps

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

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

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

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

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

Expand

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

Solution Steps

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

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

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

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

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

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