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

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

Step 1

Expand $(a + b) (b + c)$ using the FOIL Method.

Apply the distributive property.

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

Apply the distributive property.

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

Apply the distributive property.

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

Step 2

Multiply $b$ by $b$ .

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

Step 3

Expand $(a b + a c + b^{2} + b c) (c + a)$ by multiplying each term in the first expression by each term in the second expression.

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

Step 4

Simplify each term.

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

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

Multiply $a$ by $a$ .

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

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

Move $c$ .

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

Multiply $c$ by $c$ .

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

Multiply $a$ by $a$ by adding the exponents.

Move $a$ .

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

Multiply $a$ by $a$ .

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

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

Move $c$ .

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

Multiply $c$ by $c$ .

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

Step 5

Add $a b c$ and $b c a$ .

Reorder $b$ and $a$ .

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

Add $a b c$ and $a b c$ .

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