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

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

Evaluate

$$2bc$$

Solution Steps

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

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

Combine ac and -ca to get 0.

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

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

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

Combine ac and -ca to get 0.

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

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

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

Combine ba and -ab to get 0.

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

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

$$bc-c^{2}-\left(-cb\right)-\left(-c^{2}\right)$$

The opposite of -cb is cb.

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

Combine bc and cb to get 2bc.

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

The opposite of -c^{2} is c^{2}.

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

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

$$2bc$$

Expand

$$2bc$$

Solution Steps

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

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

Combine ac and -ca to get 0.

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

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

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

Combine ac and -ca to get 0.

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

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

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

Combine ba and -ab to get 0.

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

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

$$bc-c^{2}-\left(-cb\right)-\left(-c^{2}\right)$$

The opposite of -cb is cb.

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

Combine bc and cb to get 2bc.

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

The opposite of -c^{2} is c^{2}.

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

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

$$2bc$$