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

$$( a + b + c ) - ( a - b - c ) \cdot ( 2 a + b ) - ( b + c ) \cdot ( a + b )$$

Evaluate

$$c+b+a+ac-2a^{2}$$

Solution Steps

Apply the distributive property by multiplying each term of a-b-c by each term of 2a+b.

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

Combine ab and -2ba to get -ab.

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

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

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

The opposite of -ab is ab.

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

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

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

The opposite of -2ca is 2ca.

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

The opposite of -cb is cb.

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

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

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

To find the opposite of ba+b^{2}+ca+cb, find the opposite of each term.

$$a+b+c-2a^{2}+ab+b^{2}+2ca+cb-ba-b^{2}-ca-cb$$

Combine ab and -ba to get 0.

$$a+b+c-2a^{2}+b^{2}+2ca+cb-b^{2}-ca-cb$$

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

$$a+b+c-2a^{2}+2ca+cb-ca-cb$$

Combine 2ca and -ca to get ca.

$$a+b+c-2a^{2}+ca+cb-cb$$

Combine cb and -cb to get 0.

$$a+b+c-2a^{2}+ca$$

Expand

$$c+b+a+ac-2a^{2}$$

Solution Steps

Apply the distributive property by multiplying each term of a-b-c by each term of 2a+b.

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

Combine ab and -2ba to get -ab.

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

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

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

The opposite of -ab is ab.

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

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

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

The opposite of -2ca is 2ca.

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

The opposite of -cb is cb.

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

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

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

To find the opposite of ba+b^{2}+ca+cb, find the opposite of each term.

$$a+b+c-2a^{2}+ab+b^{2}+2ca+cb-ba-b^{2}-ca-cb$$

Combine ab and -ba to get 0.

$$a+b+c-2a^{2}+b^{2}+2ca+cb-b^{2}-ca-cb$$

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

$$a+b+c-2a^{2}+2ca+cb-ca-cb$$

Combine 2ca and -ca to get ca.

$$a+b+c-2a^{2}+ca+cb-cb$$

Combine cb and -cb to get 0.

$$a+b+c-2a^{2}+ca$$