Find the Product (a+2b+3c)^2

${(a + 2 b + 3 c)}^{2}$

Rewrite ${(a + 2 b + 3 c)}^{2}$ as $(a + 2 b + 3 c) (a + 2 b + 3 c)$ .

$(a + 2 b + 3 c) (a + 2 b + 3 c)$

Expand $(a + 2 b + 3 c) (a + 2 b + 3 c)$ by multiplying each term in the first expression by each term in the second expression.

$a \cdot a + a (2 b) + a (3 c) + 2 b a + 2 b (2 b) + 2 b (3 c) + 3 c a + 3 c (2 b) + 3 c (3 c)$

Simplify each term.

Multiply $a$ by $a$ .

$a^{2} + a (2 b) + a (3 c) + 2 b a + 2 b (2 b) + 2 b (3 c) + 3 c a + 3 c (2 b) + 3 c (3 c)$

Rewrite using the commutative property of multiplication.

$a^{2} + 2 a b + a (3 c) + 2 b a + 2 b (2 b) + 2 b (3 c) + 3 c a + 3 c (2 b) + 3 c (3 c)$

Rewrite using the commutative property of multiplication.

$a^{2} + 2 a b + 3 a c + 2 b a + 2 b (2 b) + 2 b (3 c) + 3 c a + 3 c (2 b) + 3 c (3 c)$

Rewrite using the commutative property of multiplication.

$a^{2} + 2 a b + 3 a c + 2 b a + 2 \cdot 2 b \cdot b + 2 b (3 c) + 3 c a + 3 c (2 b) + 3 c (3 c)$

Multiply $b$ by $b$ by adding the exponents.

Move $b$ .

$a^{2} + 2 a b + 3 a c + 2 b a + 2 \cdot 2 (b \cdot b) + 2 b (3 c) + 3 c a + 3 c (2 b) + 3 c (3 c)$

Multiply $b$ by $b$ .

$a^{2} + 2 a b + 3 a c + 2 b a + 2 \cdot 2 b^{2} + 2 b (3 c) + 3 c a + 3 c (2 b) + 3 c (3 c)$

Multiply $2$ by $2$ .

$a^{2} + 2 a b + 3 a c + 2 b a + 4 b^{2} + 2 b (3 c) + 3 c a + 3 c (2 b) + 3 c (3 c)$

Rewrite using the commutative property of multiplication.

$a^{2} + 2 a b + 3 a c + 2 b a + 4 b^{2} + 2 \cdot 3 b c + 3 c a + 3 c (2 b) + 3 c (3 c)$

Multiply $2$ by $3$ .

$a^{2} + 2 a b + 3 a c + 2 b a + 4 b^{2} + 6 b c + 3 c a + 3 c (2 b) + 3 c (3 c)$

Rewrite using the commutative property of multiplication.

$a^{2} + 2 a b + 3 a c + 2 b a + 4 b^{2} + 6 b c + 3 c a + 3 \cdot 2 c b + 3 c (3 c)$

Multiply $3$ by $2$ .

$a^{2} + 2 a b + 3 a c + 2 b a + 4 b^{2} + 6 b c + 3 c a + 6 c b + 3 c (3 c)$

Rewrite using the commutative property of multiplication.

$a^{2} + 2 a b + 3 a c + 2 b a + 4 b^{2} + 6 b c + 3 c a + 6 c b + 3 \cdot 3 c \cdot c$

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

Move $c$ .

$a^{2} + 2 a b + 3 a c + 2 b a + 4 b^{2} + 6 b c + 3 c a + 6 c b + 3 \cdot 3 (c \cdot c)$

Multiply $c$ by $c$ .

$a^{2} + 2 a b + 3 a c + 2 b a + 4 b^{2} + 6 b c + 3 c a + 6 c b + 3 \cdot 3 c^{2}$

Multiply $3$ by $3$ .

$a^{2} + 2 a b + 3 a c + 2 b a + 4 b^{2} + 6 b c + 3 c a + 6 c b + 9 c^{2}$

Add $2 a b$ and $2 b a$ .

Move $b$ .

$a^{2} + 2 a b + 2 a b + 3 a c + 4 b^{2} + 6 b c + 3 c a + 6 c b + 9 c^{2}$

Add $2 a b$ and $2 a b$ .

$a^{2} + 4 a b + 3 a c + 4 b^{2} + 6 b c + 3 c a + 6 c b + 9 c^{2}$

Add $3 a c$ and $3 c a$ .

Move $c$ .

$a^{2} + 4 a b + 4 b^{2} + 6 b c + 3 a c + 3 a c + 6 c b + 9 c^{2}$

Add $3 a c$ and $3 a c$ .

$a^{2} + 4 a b + 4 b^{2} + 6 b c + 6 a c + 6 c b + 9 c^{2}$

Add $6 b c$ and $6 c b$ .

Move $c$ .

$a^{2} + 4 a b + 4 b^{2} + 6 b c + 6 b c + 6 a c + 9 c^{2}$

Add $6 b c$ and $6 b c$ .

$a^{2} + 4 a b + 4 b^{2} + 12 b c + 6 a c + 9 c^{2}$

Find the Product (a+2b+3c)^2

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