Simplify -2d(d^3c^2-4dc^2+2d^2c)+c^2(dc^2-3d^4)

$- 2 d (d^{3} c^{2} - 4 d c^{2} + 2 d^{2} c) + c^{2} (d c^{2} - 3 d^{4})$

Simplify each term.

Apply the distributive property.

$- 2 d (d^{3} c^{2}) - 2 d (- 4 d c^{2}) - 2 d (2 d^{2} c) + c^{2} (d c^{2} - 3 d^{4})$

Simplify.

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

Move $d^{3}$ .

$- 2 (d^{3} d) c^{2} - 2 d (- 4 d c^{2}) - 2 d (2 d^{2} c) + c^{2} (d c^{2} - 3 d^{4})$

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

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

$- 2 (d^{3} d^{1}) c^{2} - 2 d (- 4 d c^{2}) - 2 d (2 d^{2} c) + c^{2} (d c^{2} - 3 d^{4})$

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

$- 2 d^{3 + 1} c^{2} - 2 d (- 4 d c^{2}) - 2 d (2 d^{2} c) + c^{2} (d c^{2} - 3 d^{4})$

Add $3$ and $1$ .

$- 2 d^{4} c^{2} - 2 d (- 4 d c^{2}) - 2 d (2 d^{2} c) + c^{2} (d c^{2} - 3 d^{4})$

Multiply $d$ by $d$ by adding the exponents.

Move $d$ .

$- 2 d^{4} c^{2} - 2 (d \cdot d) (- 4 c^{2}) - 2 d (2 d^{2} c) + c^{2} (d c^{2} - 3 d^{4})$

Multiply $d$ by $d$ .

$- 2 d^{4} c^{2} - 2 d^{2} (- 4 c^{2}) - 2 d (2 d^{2} c) + c^{2} (d c^{2} - 3 d^{4})$

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

Move $d^{2}$ .

$- 2 d^{4} c^{2} - 2 d^{2} (- 4 c^{2}) - 2 (d^{2} d) (2 c) + c^{2} (d c^{2} - 3 d^{4})$

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

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

$- 2 d^{4} c^{2} - 2 d^{2} (- 4 c^{2}) - 2 (d^{2} d^{1}) (2 c) + c^{2} (d c^{2} - 3 d^{4})$

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

$- 2 d^{4} c^{2} - 2 d^{2} (- 4 c^{2}) - 2 d^{2 + 1} (2 c) + c^{2} (d c^{2} - 3 d^{4})$

Add $2$ and $1$ .

$- 2 d^{4} c^{2} - 2 d^{2} (- 4 c^{2}) - 2 d^{3} (2 c) + c^{2} (d c^{2} - 3 d^{4})$

Simplify each term.

Rewrite using the commutative property of multiplication.

$- 2 d^{4} c^{2} - 2 \cdot - 4 d^{2} c^{2} - 2 d^{3} (2 c) + c^{2} (d c^{2} - 3 d^{4})$

Multiply $- 2$ by $- 4$ .

$- 2 d^{4} c^{2} + 8 d^{2} c^{2} - 2 d^{3} (2 c) + c^{2} (d c^{2} - 3 d^{4})$

Rewrite using the commutative property of multiplication.

$- 2 d^{4} c^{2} + 8 d^{2} c^{2} - 2 \cdot 2 d^{3} c + c^{2} (d c^{2} - 3 d^{4})$

Multiply $- 2$ by $2$ .

$- 2 d^{4} c^{2} + 8 d^{2} c^{2} - 4 d^{3} c + c^{2} (d c^{2} - 3 d^{4})$

Apply the distributive property.

$- 2 d^{4} c^{2} + 8 d^{2} c^{2} - 4 d^{3} c + c^{2} (d c^{2}) + c^{2} (- 3 d^{4})$

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

Move $c^{2}$ .

$- 2 d^{4} c^{2} + 8 d^{2} c^{2} - 4 d^{3} c + c^{2} c^{2} d + c^{2} (- 3 d^{4})$

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

$- 2 d^{4} c^{2} + 8 d^{2} c^{2} - 4 d^{3} c + c^{2 + 2} d + c^{2} (- 3 d^{4})$

Add $2$ and $2$ .

$- 2 d^{4} c^{2} + 8 d^{2} c^{2} - 4 d^{3} c + c^{4} d + c^{2} (- 3 d^{4})$

Rewrite using the commutative property of multiplication.

$- 2 d^{4} c^{2} + 8 d^{2} c^{2} - 4 d^{3} c + c^{4} d - 3 c^{2} d^{4}$

Subtract $3 c^{2} d^{4}$ from $- 2 d^{4} c^{2}$ .

Move $d^{4}$ .

$- 2 c^{2} d^{4} - 3 c^{2} d^{4} + 8 d^{2} c^{2} - 4 d^{3} c + c^{4} d$

Subtract $3 c^{2} d^{4}$ from $- 2 c^{2} d^{4}$ .

$- 5 c^{2} d^{4} + 8 d^{2} c^{2} - 4 d^{3} c + c^{4} d$

Simplify -2d(d^3c^2-4dc^2+2d^2c)+c^2(dc^2-3d^4)

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