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

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

Apply the distributive property.

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

Simplify.

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Simplify each term.

Rewrite using the commutative property of multiplication.

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

Multiply $3$ by $2$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $3$ by $- 3$ .

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

Rewrite using the commutative property of multiplication.

$6 x^{3} y - 9 x^{3} y^{2} + 3 \cdot 4 x^{4} y^{3}$

Multiply $3$ by $4$ .

$6 x^{3} y - 9 x^{3} y^{2} + 12 x^{4} y^{3}$