Find the Product 10m^4n(-2mn^3+3n)

$10 m^{4} n (- 2 m n^{3} + 3 n)$

Apply the distributive property.

$10 m^{4} n (- 2 m n^{3}) + 10 m^{4} n (3 n)$

Multiply $m^{4}$ by $m$ by adding the exponents.

Move $m$ .

$10 (m \cdot m^{4}) n (- 2 n^{3}) + 10 m^{4} n (3 n)$

Multiply $m$ by $m^{4}$ .

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

$10 (m^{1} m^{4}) n (- 2 n^{3}) + 10 m^{4} n (3 n)$

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

$10 m^{1 + 4} n (- 2 n^{3}) + 10 m^{4} n (3 n)$

Add $1$ and $4$ .

$10 m^{5} n (- 2 n^{3}) + 10 m^{4} n (3 n)$

Multiply $n$ by $n$ by adding the exponents.

Move $n$ .

$10 m^{5} n (- 2 n^{3}) + 10 m^{4} (n \cdot n) \cdot 3$

Multiply $n$ by $n$ .

$10 m^{5} n (- 2 n^{3}) + 10 m^{4} n^{2} \cdot 3$

Simplify each term.

Multiply $n$ by $n^{3}$ by adding the exponents.

Move $n^{3}$ .

$10 m^{5} (n^{3} n) \cdot - 2 + 10 m^{4} n^{2} \cdot 3$

Multiply $n^{3}$ by $n$ .

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

$10 m^{5} (n^{3} n^{1}) \cdot - 2 + 10 m^{4} n^{2} \cdot 3$

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

$10 m^{5} n^{3 + 1} \cdot - 2 + 10 m^{4} n^{2} \cdot 3$

Add $3$ and $1$ .

$10 m^{5} n^{4} \cdot - 2 + 10 m^{4} n^{2} \cdot 3$

Multiply $- 2$ by $10$ .

$- 20 m^{5} n^{4} + 10 m^{4} n^{2} \cdot 3$

Multiply $3$ by $10$ .

$- 20 m^{5} n^{4} + 30 m^{4} n^{2}$