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

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

Evaluate

$$4b^{2}+c^{2}$$

Solution Steps

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

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

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

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

Combine -b^{2} and b^{2} to get 0.

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

Combine -2bc and -2bc to get -4bc.

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

Combine -c^{2} and c^{2} to get 0.

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

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

$$-4bc+\left(2b+c\right)^{2}$$

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

$$-4bc+4b^{2}+4bc+c^{2}$$

Combine -4bc and 4bc to get 0.

$$4b^{2}+c^{2}$$

Expand

$$4b^{2}+c^{2}$$

Solution Steps

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

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

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

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

Combine -b^{2} and b^{2} to get 0.

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

Combine -2bc and -2bc to get -4bc.

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

Combine -c^{2} and c^{2} to get 0.

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

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

$$-4bc+\left(2b+c\right)^{2}$$

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

$$-4bc+4b^{2}+4bc+c^{2}$$

Combine -4bc and 4bc to get 0.

$$4b^{2}+c^{2}$$