Find the Product 5r^2s^2(-2r^2+3rs-4s^2)

$5 r^{2} s^{2} (- 2 r^{2} + 3 r s - 4 s^{2})$

Apply the distributive property.

$5 r^{2} s^{2} (- 2 r^{2}) + 5 r^{2} s^{2} (3 r s) + 5 r^{2} s^{2} (- 4 s^{2})$

Simplify.

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

Move $r^{2}$ .

$5 (r^{2} r^{2}) s^{2} \cdot - 2 + 5 r^{2} s^{2} (3 r s) + 5 r^{2} s^{2} (- 4 s^{2})$

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

$5 r^{2 + 2} s^{2} \cdot - 2 + 5 r^{2} s^{2} (3 r s) + 5 r^{2} s^{2} (- 4 s^{2})$

Add $2$ and $2$ .

$5 r^{4} s^{2} \cdot - 2 + 5 r^{2} s^{2} (3 r s) + 5 r^{2} s^{2} (- 4 s^{2})$

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

Move $r$ .

$5 r^{4} s^{2} \cdot - 2 + 5 (r \cdot r^{2}) s^{2} (3 s) + 5 r^{2} s^{2} (- 4 s^{2})$

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

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

$5 r^{4} s^{2} \cdot - 2 + 5 (r^{1} r^{2}) s^{2} (3 s) + 5 r^{2} s^{2} (- 4 s^{2})$

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

$5 r^{4} s^{2} \cdot - 2 + 5 r^{1 + 2} s^{2} (3 s) + 5 r^{2} s^{2} (- 4 s^{2})$

Add $1$ and $2$ .

$5 r^{4} s^{2} \cdot - 2 + 5 r^{3} s^{2} (3 s) + 5 r^{2} s^{2} (- 4 s^{2})$

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

Move $s^{2}$ .

$5 r^{4} s^{2} \cdot - 2 + 5 r^{3} s^{2} (3 s) + 5 r^{2} (s^{2} s^{2}) \cdot - 4$

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

$5 r^{4} s^{2} \cdot - 2 + 5 r^{3} s^{2} (3 s) + 5 r^{2} s^{2 + 2} \cdot - 4$

Add $2$ and $2$ .

$5 r^{4} s^{2} \cdot - 2 + 5 r^{3} s^{2} (3 s) + 5 r^{2} s^{4} \cdot - 4$

Simplify each term.

Multiply $- 2$ by $5$ .

$- 10 r^{4} s^{2} + 5 r^{3} s^{2} (3 s) + 5 r^{2} s^{4} \cdot - 4$

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

Move $s$ .

$- 10 r^{4} s^{2} + 5 r^{3} (s \cdot s^{2}) \cdot 3 + 5 r^{2} s^{4} \cdot - 4$

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

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

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

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

$- 10 r^{4} s^{2} + 5 r^{3} s^{1 + 2} \cdot 3 + 5 r^{2} s^{4} \cdot - 4$

Add $1$ and $2$ .

$- 10 r^{4} s^{2} + 5 r^{3} s^{3} \cdot 3 + 5 r^{2} s^{4} \cdot - 4$

Multiply $3$ by $5$ .

$- 10 r^{4} s^{2} + 15 r^{3} s^{3} + 5 r^{2} s^{4} \cdot - 4$

Multiply $- 4$ by $5$ .

$- 10 r^{4} s^{2} + 15 r^{3} s^{3} - 20 r^{2} s^{4}$