Simplify (4n^2-6np+9p^2)(2n+3p)

$(4 n^{2} - 6 n p + 9 p^{2}) (2 n + 3 p)$

Step 1

Expand $(4 n^{2} - 6 n p + 9 p^{2}) (2 n + 3 p)$ by multiplying each term in the first expression by each term in the second expression.

$4 n^{2} (2 n) + 4 n^{2} (3 p) - 6 n p (2 n) - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

Step 2

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$4 \cdot 2 n^{2} n + 4 n^{2} (3 p) - 6 n p (2 n) - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

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

Move $n$ .

$4 \cdot 2 (n \cdot n^{2}) + 4 n^{2} (3 p) - 6 n p (2 n) - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

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

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

$4 \cdot 2 (n^{1} n^{2}) + 4 n^{2} (3 p) - 6 n p (2 n) - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

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

$4 \cdot 2 n^{1 + 2} + 4 n^{2} (3 p) - 6 n p (2 n) - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

Add $1$ and $2$ .

$4 \cdot 2 n^{3} + 4 n^{2} (3 p) - 6 n p (2 n) - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

Multiply $4$ by $2$ .

$8 n^{3} + 4 n^{2} (3 p) - 6 n p (2 n) - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

Rewrite using the commutative property of multiplication.

$8 n^{3} + 4 \cdot 3 n^{2} p - 6 n p (2 n) - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

Multiply $4$ by $3$ .

$8 n^{3} + 12 n^{2} p - 6 n p (2 n) - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

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

Move $n$ .

$8 n^{3} + 12 n^{2} p - 6 (n \cdot n) p \cdot 2 - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

Multiply $n$ by $n$ .

$8 n^{3} + 12 n^{2} p - 6 n^{2} p \cdot 2 - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

Multiply $2$ by $- 6$ .

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 6 n p (3 p) + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

Multiply $p$ by $p$ by adding the exponents.

Move $p$ .

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 6 n (p \cdot p) \cdot 3 + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

Multiply $p$ by $p$ .

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 6 n p^{2} \cdot 3 + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

Multiply $3$ by $- 6$ .

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 18 n p^{2} + 9 p^{2} (2 n) + 9 p^{2} (3 p)$

Rewrite using the commutative property of multiplication.

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 18 n p^{2} + 9 \cdot 2 p^{2} n + 9 p^{2} (3 p)$

Multiply $9$ by $2$ .

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 18 n p^{2} + 18 p^{2} n + 9 p^{2} (3 p)$

Rewrite using the commutative property of multiplication.

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 18 n p^{2} + 18 p^{2} n + 9 \cdot 3 p^{2} p$

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

Move $p$ .

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 18 n p^{2} + 18 p^{2} n + 9 \cdot 3 (p \cdot p^{2})$

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

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

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 18 n p^{2} + 18 p^{2} n + 9 \cdot 3 (p^{1} p^{2})$

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

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 18 n p^{2} + 18 p^{2} n + 9 \cdot 3 p^{1 + 2}$

Add $1$ and $2$ .

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 18 n p^{2} + 18 p^{2} n + 9 \cdot 3 p^{3}$

Multiply $9$ by $3$ .

$8 n^{3} + 12 n^{2} p - 12 n^{2} p - 18 n p^{2} + 18 p^{2} n + 27 p^{3}$

Combine the opposite terms in $8 n^{3} + 12 n^{2} p - 12 n^{2} p - 18 n p^{2} + 18 p^{2} n + 27 p^{3}$ .

Subtract $12 n^{2} p$ from $12 n^{2} p$ .

$8 n^{3} + 0 - 18 n p^{2} + 18 p^{2} n + 27 p^{3}$

Add $8 n^{3}$ and $0$ .

$8 n^{3} - 18 n p^{2} + 18 p^{2} n + 27 p^{3}$

Reorder the factors in the terms $- 18 n p^{2}$ and $18 p^{2} n$ .

$8 n^{3} - 18 p^{2} n + 18 p^{2} n + 27 p^{3}$

Add $- 18 p^{2} n$ and $18 p^{2} n$ .

$8 n^{3} + 0 + 27 p^{3}$

Add $8 n^{3}$ and $0$ .

$8 n^{3} + 27 p^{3}$

Simplify (4n^2-6np+9p^2)(2n+3p)

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