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

$$( a + b + c ) ( - a + b + c ) + ( a - b + c ) ( a + b - c )$$

Evaluate

$$4bc$$

Solution Steps

Apply the distributive property by multiplying each term of a+b+c by each term of -a+b+c.

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

Combine bc and cb to get 2bc.

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

Apply the distributive property by multiplying each term of a-b+c by each term of a+b-c.

$$a\left(-a\right)+ab+ac+b\left(-a\right)+b^{2}+2bc+c\left(-a\right)+c^{2}+a^{2}+ab-ac-ba-b^{2}+bc+ca+cb-c^{2}$$

Combine ab and -ba to get 0.

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

Combine -ac and ca to get 0.

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

Combine bc and cb to get 2bc.

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

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

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

Combine 2bc and 2bc to get 4bc.

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

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

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

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

$$a^{2}\left(-1\right)+ab+ac+b\left(-1\right)a+4bc+c\left(-1\right)a+a^{2}$$

Combine ab and b\left(-1\right)a to get 0.

$$a^{2}\left(-1\right)+ac+4bc+c\left(-1\right)a+a^{2}$$

Combine ac and c\left(-1\right)a to get 0.

$$a^{2}\left(-1\right)+4bc+a^{2}$$

Combine a^{2}\left(-1\right) and a^{2} to get 0.

$$4bc$$

Expand

$$4bc$$

Solution Steps

Apply the distributive property by multiplying each term of a+b+c by each term of -a+b+c.

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

Combine bc and cb to get 2bc.

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

Apply the distributive property by multiplying each term of a-b+c by each term of a+b-c.

$$a\left(-a\right)+ab+ac+b\left(-a\right)+b^{2}+2bc+c\left(-a\right)+c^{2}+a^{2}+ab-ac-ba-b^{2}+bc+ca+cb-c^{2}$$

Combine ab and -ba to get 0.

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

Combine -ac and ca to get 0.

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

Combine bc and cb to get 2bc.

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

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

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

Combine 2bc and 2bc to get 4bc.

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

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

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

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

$$a^{2}\left(-1\right)+ab+ac+b\left(-1\right)a+4bc+c\left(-1\right)a+a^{2}$$

Combine ab and b\left(-1\right)a to get 0.

$$a^{2}\left(-1\right)+ac+4bc+c\left(-1\right)a+a^{2}$$

Combine ac and c\left(-1\right)a to get 0.

$$a^{2}\left(-1\right)+4bc+a^{2}$$

Combine a^{2}\left(-1\right) and a^{2} to get 0.

$$4bc$$