Expand the Trigonometric Expression (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.

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

Step 2

Simplify terms.

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

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

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

Subtract $a c$ from $a c$ .

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

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

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

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

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

Subtract $b c$ from $b c$ .

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

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

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

Simplify each term.

Multiply $a$ by $a$ .

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

Multiply $b$ by $b$ .

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

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 $a b$ .

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