Find the Product (2x+5y)(3x^2-4xy+2y^2)

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

Step 1

Expand $(2 x + 5 y) (3 x^{2} - 4 x y + 2 y^{2})$ by multiplying each term in the first expression by each term in the second expression.

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

Step 2

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x^{2}$ .

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

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

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

$2 \cdot 3 (x^{2} x^{1}) + 2 x (- 4 x y) + 2 x (2 y^{2}) + 5 y (3 x^{2}) + 5 y (- 4 x y) + 5 y (2 y^{2})$

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

$2 \cdot 3 x^{2 + 1} + 2 x (- 4 x y) + 2 x (2 y^{2}) + 5 y (3 x^{2}) + 5 y (- 4 x y) + 5 y (2 y^{2})$

Add $2$ and $1$ .

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

Multiply $2$ by $3$ .

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

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

Move $x$ .

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

Multiply $x$ by $x$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $2$ by $- 4$ .

$6 x^{3} - 8 x^{2} y + 2 x (2 y^{2}) + 5 y (3 x^{2}) + 5 y (- 4 x y) + 5 y (2 y^{2})$

Rewrite using the commutative property of multiplication.

$6 x^{3} - 8 x^{2} y + 2 \cdot 2 x y^{2} + 5 y (3 x^{2}) + 5 y (- 4 x y) + 5 y (2 y^{2})$

Multiply $2$ by $2$ .

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 5 y (3 x^{2}) + 5 y (- 4 x y) + 5 y (2 y^{2})$

Rewrite using the commutative property of multiplication.

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 5 \cdot 3 y x^{2} + 5 y (- 4 x y) + 5 y (2 y^{2})$

Multiply $5$ by $3$ .

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 15 y x^{2} + 5 y (- 4 x y) + 5 y (2 y^{2})$

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

Move $y$ .

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 15 y x^{2} + 5 (y \cdot y) (- 4 x) + 5 y (2 y^{2})$

Multiply $y$ by $y$ .

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 15 y x^{2} + 5 y^{2} (- 4 x) + 5 y (2 y^{2})$

Rewrite using the commutative property of multiplication.

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 15 y x^{2} + 5 \cdot - 4 y^{2} x + 5 y (2 y^{2})$

Multiply $5$ by $- 4$ .

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 15 y x^{2} - 20 y^{2} x + 5 y (2 y^{2})$

Rewrite using the commutative property of multiplication.

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 15 y x^{2} - 20 y^{2} x + 5 \cdot 2 y \cdot y^{2}$

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

Move $y^{2}$ .

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 15 y x^{2} - 20 y^{2} x + 5 \cdot 2 (y^{2} y)$

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

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

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 15 y x^{2} - 20 y^{2} x + 5 \cdot 2 (y^{2} y^{1})$

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

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 15 y x^{2} - 20 y^{2} x + 5 \cdot 2 y^{2 + 1}$

Add $2$ and $1$ .

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 15 y x^{2} - 20 y^{2} x + 5 \cdot 2 y^{3}$

Multiply $5$ by $2$ .

$6 x^{3} - 8 x^{2} y + 4 x y^{2} + 15 y x^{2} - 20 y^{2} x + 10 y^{3}$

Step 3

Add $- 8 x^{2} y$ and $15 y x^{2}$ .

Move $y$ .

$6 x^{3} - 8 x^{2} y + 15 x^{2} y + 4 x y^{2} - 20 y^{2} x + 10 y^{3}$

Add $- 8 x^{2} y$ and $15 x^{2} y$ .

$6 x^{3} + 7 x^{2} y + 4 x y^{2} - 20 y^{2} x + 10 y^{3}$

Step 4

Subtract $20 y^{2} x$ from $4 x y^{2}$ .

Move $y^{2}$ .

$6 x^{3} + 7 x^{2} y + 4 x y^{2} - 20 x y^{2} + 10 y^{3}$

Subtract $20 x y^{2}$ from $4 x y^{2}$ .

$6 x^{3} + 7 x^{2} y - 16 x y^{2} + 10 y^{3}$

Find the Product (2x+5y)(3x^2-4xy+2y^2)

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