Simplify (2x+y)(3x^2+2xy+5y^2)

$(2 x + y) (3 x^{2} + 2 x y + 5 y^{2})$

Step 1

Expand $(2 x + y) (3 x^{2} + 2 x y + 5 y^{2})$ by multiplying each term in the first expression by each term in the second expression.

$2 x (3 x^{2}) + 2 x (2 x y) + 2 x (5 y^{2}) + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Step 2

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 \cdot 3 x \cdot x^{2} + 2 x (2 x y) + 2 x (5 y^{2}) + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Multiply $x$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$2 \cdot 3 (x^{2} x) + 2 x (2 x y) + 2 x (5 y^{2}) + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Multiply $x^{2}$ by $x$ .

Raise $x$ to the power of $1$ .

$2 \cdot 3 (x^{2} x^{1}) + 2 x (2 x y) + 2 x (5 y^{2}) + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$2 \cdot 3 x^{2 + 1} + 2 x (2 x y) + 2 x (5 y^{2}) + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Add $2$ and $1$ .

$2 \cdot 3 x^{3} + 2 x (2 x y) + 2 x (5 y^{2}) + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Multiply $2$ by $3$ .

$6 x^{3} + 2 x (2 x y) + 2 x (5 y^{2}) + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

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

Move $x$ .

$6 x^{3} + 2 (x \cdot x) (2 y) + 2 x (5 y^{2}) + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Multiply $x$ by $x$ .

$6 x^{3} + 2 x^{2} (2 y) + 2 x (5 y^{2}) + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Rewrite using the commutative property of multiplication.

$6 x^{3} + 2 \cdot 2 x^{2} y + 2 x (5 y^{2}) + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Multiply $2$ by $2$ .

$6 x^{3} + 4 x^{2} y + 2 x (5 y^{2}) + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Rewrite using the commutative property of multiplication.

$6 x^{3} + 4 x^{2} y + 2 \cdot 5 x y^{2} + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Multiply $2$ by $5$ .

$6 x^{3} + 4 x^{2} y + 10 x y^{2} + y (3 x^{2}) + y (2 x y) + y (5 y^{2})$

Rewrite using the commutative property of multiplication.

$6 x^{3} + 4 x^{2} y + 10 x y^{2} + 3 y x^{2} + y (2 x y) + y (5 y^{2})$

Rewrite using the commutative property of multiplication.

$6 x^{3} + 4 x^{2} y + 10 x y^{2} + 3 y x^{2} + 2 y (x y) + y (5 y^{2})$

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

Move $y$ .

$6 x^{3} + 4 x^{2} y + 10 x y^{2} + 3 y x^{2} + 2 (y \cdot y) x + y (5 y^{2})$

Multiply $y$ by $y$ .

$6 x^{3} + 4 x^{2} y + 10 x y^{2} + 3 y x^{2} + 2 y^{2} x + y (5 y^{2})$

Rewrite using the commutative property of multiplication.

$6 x^{3} + 4 x^{2} y + 10 x y^{2} + 3 y x^{2} + 2 y^{2} x + 5 y \cdot y^{2}$

Multiply $y$ by $y^{2}$ by adding the exponents.

Move $y^{2}$ .

$6 x^{3} + 4 x^{2} y + 10 x y^{2} + 3 y x^{2} + 2 y^{2} x + 5 (y^{2} y)$

Multiply $y^{2}$ by $y$ .

Raise $y$ to the power of $1$ .

$6 x^{3} + 4 x^{2} y + 10 x y^{2} + 3 y x^{2} + 2 y^{2} x + 5 (y^{2} y^{1})$

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$6 x^{3} + 4 x^{2} y + 10 x y^{2} + 3 y x^{2} + 2 y^{2} x + 5 y^{2 + 1}$

Add $2$ and $1$ .

$6 x^{3} + 4 x^{2} y + 10 x y^{2} + 3 y x^{2} + 2 y^{2} x + 5 y^{3}$

Step 3

Add $4 x^{2} y$ and $3 y x^{2}$ .

Move $y$ .

$6 x^{3} + 4 x^{2} y + 3 x^{2} y + 10 x y^{2} + 2 y^{2} x + 5 y^{3}$

Add $4 x^{2} y$ and $3 x^{2} y$ .

$6 x^{3} + 7 x^{2} y + 10 x y^{2} + 2 y^{2} x + 5 y^{3}$

Step 4

Add $10 x y^{2}$ and $2 y^{2} x$ .

Move $y^{2}$ .

$6 x^{3} + 7 x^{2} y + 10 x y^{2} + 2 x y^{2} + 5 y^{3}$

Add $10 x y^{2}$ and $2 x y^{2}$ .

$6 x^{3} + 7 x^{2} y + 12 x y^{2} + 5 y^{3}$

Simplify (2x+y)(3x^2+2xy+5y^2)

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