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

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

Evaluate

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

Solution Steps

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

$$\left(-a\right)a-\left(-a\right)b+\left(-a\right)c+\left(-a\right)d+ba-b^{2}+bc+bd+ca-cb+c^{2}+cd+da-db+dc+d^{2}-\left(a+b-c+d\right)\left(a+b+c-d\right)$$

Multiply -1 and -1 to get 1.

$$\left(-a\right)a+ab+\left(-a\right)c+\left(-a\right)d+ba-b^{2}+bc+bd+ca-cb+c^{2}+cd+da-db+dc+d^{2}-\left(a+b-c+d\right)\left(a+b+c-d\right)$$

Combine ab and ba to get 2ab.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+bc+bd+ca-cb+c^{2}+cd+da-db+dc+d^{2}-\left(a+b-c+d\right)\left(a+b+c-d\right)$$

Combine bc and -cb to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+bd+ca+c^{2}+cd+da-db+dc+d^{2}-\left(a+b-c+d\right)\left(a+b+c-d\right)$$

Combine bd and -db to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+cd+da+dc+d^{2}-\left(a+b-c+d\right)\left(a+b+c-d\right)$$

Combine cd and dc to get 2cd.

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

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

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+ab+ac-ad+ba+b^{2}+bc-bd-ca-cb-c^{2}+dc+da+db+dc-d^{2}\right)$$

Combine ab and ba to get 2ab.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab+ac-ad+b^{2}+bc-bd-ca-cb-c^{2}+dc+da+db+dc-d^{2}\right)$$

Combine ac and -ca to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab-ad+b^{2}+bc-bd-cb-c^{2}+dc+da+db+dc-d^{2}\right)$$

Combine bc and -cb to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab-ad+b^{2}-bd-c^{2}+dc+da+db+dc-d^{2}\right)$$

Combine -ad and da to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab+b^{2}-bd-c^{2}+dc+db+dc-d^{2}\right)$$

Combine -bd and db to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab+b^{2}-c^{2}+dc+dc-d^{2}\right)$$

Combine dc and dc to get 2dc.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab+b^{2}-c^{2}+2dc-d^{2}\right)$$

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

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-a^{2}-2ab-b^{2}-\left(-c^{2}\right)-2dc-\left(-d^{2}\right)$$

Combine 2ab and -2ab to get 0.

$$\left(-a\right)a+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-a^{2}-b^{2}-\left(-c^{2}\right)-2dc-\left(-d^{2}\right)$$

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

$$\left(-a\right)a+\left(-a\right)c+\left(-a\right)d-2b^{2}+ca+c^{2}+2cd+da+d^{2}-a^{2}-\left(-c^{2}\right)-2dc-\left(-d^{2}\right)$$

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

$$\left(-a\right)a+\left(-a\right)c+\left(-a\right)d-2b^{2}+ca+c^{2}+2cd+da+d^{2}-a^{2}+c^{2}-2dc-\left(-d^{2}\right)$$

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

$$\left(-a\right)a+\left(-a\right)c+\left(-a\right)d-2b^{2}+ca+2c^{2}+2cd+da+d^{2}-a^{2}-2dc-\left(-d^{2}\right)$$

Combine 2cd and -2dc to get 0.

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

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

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

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

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

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

$$-a^{2}-ac-ad-2b^{2}+ca+2c^{2}+da+2d^{2}-a^{2}$$

Combine -ac and ca to get 0.

$$-a^{2}-ad-2b^{2}+2c^{2}+da+2d^{2}-a^{2}$$

Combine -ad and da to get 0.

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

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

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

Expand

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

Solution Steps

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

$$\left(-a\right)a-\left(-a\right)b+\left(-a\right)c+\left(-a\right)d+ba-b^{2}+bc+bd+ca-cb+c^{2}+cd+da-db+dc+d^{2}-\left(a+b-c+d\right)\left(a+b+c-d\right)$$

Multiply -1 and -1 to get 1.

$$\left(-a\right)a+ab+\left(-a\right)c+\left(-a\right)d+ba-b^{2}+bc+bd+ca-cb+c^{2}+cd+da-db+dc+d^{2}-\left(a+b-c+d\right)\left(a+b+c-d\right)$$

Combine ab and ba to get 2ab.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+bc+bd+ca-cb+c^{2}+cd+da-db+dc+d^{2}-\left(a+b-c+d\right)\left(a+b+c-d\right)$$

Combine bc and -cb to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+bd+ca+c^{2}+cd+da-db+dc+d^{2}-\left(a+b-c+d\right)\left(a+b+c-d\right)$$

Combine bd and -db to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+cd+da+dc+d^{2}-\left(a+b-c+d\right)\left(a+b+c-d\right)$$

Combine cd and dc to get 2cd.

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

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

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+ab+ac-ad+ba+b^{2}+bc-bd-ca-cb-c^{2}+dc+da+db+dc-d^{2}\right)$$

Combine ab and ba to get 2ab.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab+ac-ad+b^{2}+bc-bd-ca-cb-c^{2}+dc+da+db+dc-d^{2}\right)$$

Combine ac and -ca to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab-ad+b^{2}+bc-bd-cb-c^{2}+dc+da+db+dc-d^{2}\right)$$

Combine bc and -cb to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab-ad+b^{2}-bd-c^{2}+dc+da+db+dc-d^{2}\right)$$

Combine -ad and da to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab+b^{2}-bd-c^{2}+dc+db+dc-d^{2}\right)$$

Combine -bd and db to get 0.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab+b^{2}-c^{2}+dc+dc-d^{2}\right)$$

Combine dc and dc to get 2dc.

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-\left(a^{2}+2ab+b^{2}-c^{2}+2dc-d^{2}\right)$$

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

$$\left(-a\right)a+2ab+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-a^{2}-2ab-b^{2}-\left(-c^{2}\right)-2dc-\left(-d^{2}\right)$$

Combine 2ab and -2ab to get 0.

$$\left(-a\right)a+\left(-a\right)c+\left(-a\right)d-b^{2}+ca+c^{2}+2cd+da+d^{2}-a^{2}-b^{2}-\left(-c^{2}\right)-2dc-\left(-d^{2}\right)$$

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

$$\left(-a\right)a+\left(-a\right)c+\left(-a\right)d-2b^{2}+ca+c^{2}+2cd+da+d^{2}-a^{2}-\left(-c^{2}\right)-2dc-\left(-d^{2}\right)$$

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

$$\left(-a\right)a+\left(-a\right)c+\left(-a\right)d-2b^{2}+ca+c^{2}+2cd+da+d^{2}-a^{2}+c^{2}-2dc-\left(-d^{2}\right)$$

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

$$\left(-a\right)a+\left(-a\right)c+\left(-a\right)d-2b^{2}+ca+2c^{2}+2cd+da+d^{2}-a^{2}-2dc-\left(-d^{2}\right)$$

Combine 2cd and -2dc to get 0.

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

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

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

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

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

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

$$-a^{2}-ac-ad-2b^{2}+ca+2c^{2}+da+2d^{2}-a^{2}$$

Combine -ac and ca to get 0.

$$-a^{2}-ad-2b^{2}+2c^{2}+da+2d^{2}-a^{2}$$

Combine -ad and da to get 0.

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

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

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