Simplify (a+b+c)^2

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

Step 1

Rewrite ${(a + b + c)}^{2}$ as $(a + b + c) (a + b + c)$ .

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

Step 2

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 3

Simplify each term.

Multiply $a$ by $a$ .

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

Multiply $b$ by $b$ .

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

Multiply $c$ by $c$ .

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

Step 4

Add $a b$ and $b a$ .

Reorder $b$ and $a$ .

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

Add $a b$ and $a b$ .

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

Step 5

Add $a c$ and $c a$ .

Reorder $c$ and $a$ .

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

Add $a c$ and $a c$ .

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

Step 6

Add $b c$ and $c b$ .

Reorder $c$ and $b$ .

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

Add $b c$ and $b c$ .

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

Step 7

Move $2 b c$ .

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

Step 8

Move $b^{2}$ .

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