Simplify (2x+3y)^2+2(2x+3y)(x+y)+(x+y)^2

${(2 x + 3 y)}^{2} + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Step 1

Simplify each term.

Rewrite ${(2 x + 3 y)}^{2}$ as $(2 x + 3 y) (2 x + 3 y)$ .

$(2 x + 3 y) (2 x + 3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Expand $(2 x + 3 y) (2 x + 3 y)$ using the FOIL Method.

Apply the distributive property.

$2 x (2 x + 3 y) + 3 y (2 x + 3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Apply the distributive property.

$2 x (2 x) + 2 x (3 y) + 3 y (2 x + 3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Apply the distributive property.

$2 x (2 x) + 2 x (3 y) + 3 y (2 x) + 3 y (3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 2 x \cdot x + 2 x (3 y) + 3 y (2 x) + 3 y (3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$2 \cdot 2 (x \cdot x) + 2 x (3 y) + 3 y (2 x) + 3 y (3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Multiply $x$ by $x$ .

$2 \cdot 2 x^{2} + 2 x (3 y) + 3 y (2 x) + 3 y (3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Multiply $2$ by $2$ .

$4 x^{2} + 2 x (3 y) + 3 y (2 x) + 3 y (3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Rewrite using the commutative property of multiplication.

$4 x^{2} + 2 \cdot 3 x y + 3 y (2 x) + 3 y (3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Multiply $2$ by $3$ .

$4 x^{2} + 6 x y + 3 y (2 x) + 3 y (3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Rewrite using the commutative property of multiplication.

$4 x^{2} + 6 x y + 3 \cdot 2 y x + 3 y (3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Multiply $3$ by $2$ .

$4 x^{2} + 6 x y + 6 y x + 3 y (3 y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Rewrite using the commutative property of multiplication.

$4 x^{2} + 6 x y + 6 y x + 3 \cdot 3 y \cdot y + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$4 x^{2} + 6 x y + 6 y x + 3 \cdot 3 (y \cdot y) + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Multiply $y$ by $y$ .

$4 x^{2} + 6 x y + 6 y x + 3 \cdot 3 y^{2} + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Multiply $3$ by $3$ .

$4 x^{2} + 6 x y + 6 y x + 9 y^{2} + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Add $6 x y$ and $6 y x$ .

Move $y$ .

$4 x^{2} + 6 x y + 6 x y + 9 y^{2} + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Add $6 x y$ and $6 x y$ .

$4 x^{2} + 12 x y + 9 y^{2} + 2 (2 x + 3 y) (x + y) + {(x + y)}^{2}$

Apply the distributive property.

$4 x^{2} + 12 x y + 9 y^{2} + (2 (2 x) + 2 (3 y)) (x + y) + {(x + y)}^{2}$

Multiply $2$ by $2$ .

$4 x^{2} + 12 x y + 9 y^{2} + (4 x + 2 (3 y)) (x + y) + {(x + y)}^{2}$

Multiply $3$ by $2$ .

$4 x^{2} + 12 x y + 9 y^{2} + (4 x + 6 y) (x + y) + {(x + y)}^{2}$

Expand $(4 x + 6 y) (x + y)$ using the FOIL Method.

Apply the distributive property.

$4 x^{2} + 12 x y + 9 y^{2} + 4 x (x + y) + 6 y (x + y) + {(x + y)}^{2}$

Apply the distributive property.

$4 x^{2} + 12 x y + 9 y^{2} + 4 x \cdot x + 4 x y + 6 y (x + y) + {(x + y)}^{2}$

Apply the distributive property.

$4 x^{2} + 12 x y + 9 y^{2} + 4 x \cdot x + 4 x y + 6 y x + 6 y \cdot y + {(x + y)}^{2}$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$4 x^{2} + 12 x y + 9 y^{2} + 4 (x \cdot x) + 4 x y + 6 y x + 6 y \cdot y + {(x + y)}^{2}$

Multiply $x$ by $x$ .

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 4 x y + 6 y x + 6 y \cdot y + {(x + y)}^{2}$

Multiply $y$ by $y$ by adding the exponents.

Move $y$ .

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 4 x y + 6 y x + 6 (y \cdot y) + {(x + y)}^{2}$

Multiply $y$ by $y$ .

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 4 x y + 6 y x + 6 y^{2} + {(x + y)}^{2}$

Add $4 x y$ and $6 y x$ .

Move $y$ .

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 4 x y + 6 x y + 6 y^{2} + {(x + y)}^{2}$

Add $4 x y$ and $6 x y$ .

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 10 x y + 6 y^{2} + {(x + y)}^{2}$

Rewrite ${(x + y)}^{2}$ as $(x + y) (x + y)$ .

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 10 x y + 6 y^{2} + (x + y) (x + y)$

Expand $(x + y) (x + y)$ using the FOIL Method.

Apply the distributive property.

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 10 x y + 6 y^{2} + x (x + y) + y (x + y)$

Apply the distributive property.

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 10 x y + 6 y^{2} + x \cdot x + x y + y (x + y)$

Apply the distributive property.

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 10 x y + 6 y^{2} + x \cdot x + x y + y x + y \cdot y$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 10 x y + 6 y^{2} + x^{2} + x y + y x + y \cdot y$

Multiply $y$ by $y$ .

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 10 x y + 6 y^{2} + x^{2} + x y + y x + y^{2}$

Add $x y$ and $y x$ .

Reorder $y$ and $x$ .

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 10 x y + 6 y^{2} + x^{2} + x y + x y + y^{2}$

Add $x y$ and $x y$ .

$4 x^{2} + 12 x y + 9 y^{2} + 4 x^{2} + 10 x y + 6 y^{2} + x^{2} + 2 x y + y^{2}$

Step 2

Simplify by adding terms.

Add $4 x^{2}$ and $4 x^{2}$ .

$8 x^{2} + 12 x y + 9 y^{2} + 10 x y + 6 y^{2} + x^{2} + 2 x y + y^{2}$

Add $8 x^{2}$ and $x^{2}$ .

$9 x^{2} + 12 x y + 9 y^{2} + 10 x y + 6 y^{2} + 2 x y + y^{2}$

Add $12 x y$ and $10 x y$ .

$9 x^{2} + 9 y^{2} + 22 x y + 6 y^{2} + 2 x y + y^{2}$

Add $9 y^{2}$ and $6 y^{2}$ .

$9 x^{2} + 15 y^{2} + 22 x y + 2 x y + y^{2}$

Add $15 y^{2}$ and $y^{2}$ .

$9 x^{2} + 16 y^{2} + 22 x y + 2 x y$

Add $22 x y$ and $2 x y$ .

$9 x^{2} + 16 y^{2} + 24 x y$

Simplify (2x+3y)^2+2(2x+3y)(x+y)+(x+y)^2

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