Simplify (a+b+c)(a+b-c)

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

Step 1

Expand $(a + b + c) (a + b - c)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify terms.

Combine the opposite terms in $a \cdot a + a b + a (- c) + b a + b \cdot b + b (- c) + c a + c b + c (- c)$ .

Reorder the factors in the terms $a (- c)$ and $c a$ .

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

Add $- a c$ and $a c$ .

$a \cdot a + a b + b a + b \cdot b + b (- c) + 0 + c b + c (- c)$

Add $a \cdot a + a b + b a + b \cdot b + b (- c)$ and $0$ .

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

Reorder the factors in the terms $b (- c)$ and $c b$ .

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

Add $- b c$ and $b c$ .

$a \cdot a + a b + b a + b \cdot b + 0 + c (- c)$

Add $a \cdot a + a b + b a + b \cdot b$ and $0$ .

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

Simplify each term.

Multiply $a$ by $a$ .

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

Multiply $b$ by $b$ .

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

Rewrite using the commutative property of multiplication.

$a^{2} + a b + b a + b^{2} - c \cdot c$

Multiply $c$ by $c$ by adding the exponents.

Move $c$ .

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

Multiply $c$ by $c$ .

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

Step 3

Add $a b$ and $b a$ .

Reorder $b$ and $a$ .

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

Add $a b$ and $a b$ .

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