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

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

Step 1

Expand

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

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

Step 2

Simplify terms.

Combine the opposite terms in

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

Reorder the factors in the terms

$a b$ and

$b (- a)$ .

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

Subtract

$a b$ from

$a b$ .

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

Add

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

$0$ .

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

Reorder the factors in the terms

$a c$ and

$c (- a)$ .

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

Subtract

$a c$ from

$a c$ .

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

Add

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

$0$ .

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

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Multiply

$a$ by

$a$ by adding the exponents.

Move

$a$ .

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

Multiply

$a$ by

$a$ .

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

Multiply

$b$ by

$b$ .

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

Multiply

$c$ by

$c$ .

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

Step 3

Add

$b c$ and

$c b$ .

Reorder

$c$ and

$b$ .

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

Add

$b c$ and

$b c$ .

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

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

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