Expand the Trigonometric Expression (n^2-3n+1)(2n+3)

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

Step 1

Expand $(n^{2} - 3 n + 1) (2 n + 3)$ by multiplying each term in the first expression by each term in the second expression.

$n^{2} (2 n) + n^{2} \cdot 3 - 3 n (2 n) - 3 n \cdot 3 + 1 (2 n) + 1 \cdot 3$

Step 2

Simplify terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$2 n^{2} n + n^{2} \cdot 3 - 3 n (2 n) - 3 n \cdot 3 + 1 (2 n) + 1 \cdot 3$

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

Move $n$ .

$2 (n \cdot n^{2}) + n^{2} \cdot 3 - 3 n (2 n) - 3 n \cdot 3 + 1 (2 n) + 1 \cdot 3$

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

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

$2 (n^{1} n^{2}) + n^{2} \cdot 3 - 3 n (2 n) - 3 n \cdot 3 + 1 (2 n) + 1 \cdot 3$

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

$2 n^{1 + 2} + n^{2} \cdot 3 - 3 n (2 n) - 3 n \cdot 3 + 1 (2 n) + 1 \cdot 3$

Add $1$ and $2$ .

$2 n^{3} + n^{2} \cdot 3 - 3 n (2 n) - 3 n \cdot 3 + 1 (2 n) + 1 \cdot 3$

Move $3$ to the left of $n^{2}$ .

$2 n^{3} + 3 \cdot n^{2} - 3 n (2 n) - 3 n \cdot 3 + 1 (2 n) + 1 \cdot 3$

Rewrite using the commutative property of multiplication.

$2 n^{3} + 3 n^{2} - 3 \cdot 2 n \cdot n - 3 n \cdot 3 + 1 (2 n) + 1 \cdot 3$

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

Move $n$ .

$2 n^{3} + 3 n^{2} - 3 \cdot 2 (n \cdot n) - 3 n \cdot 3 + 1 (2 n) + 1 \cdot 3$

Multiply $n$ by $n$ .

$2 n^{3} + 3 n^{2} - 3 \cdot 2 n^{2} - 3 n \cdot 3 + 1 (2 n) + 1 \cdot 3$

Multiply $- 3$ by $2$ .

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

Multiply $3$ by $- 3$ .

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

Multiply $2 n$ by $1$ .

$2 n^{3} + 3 n^{2} - 6 n^{2} - 9 n + 2 n + 1 \cdot 3$

Multiply $3$ by $1$ .

$2 n^{3} + 3 n^{2} - 6 n^{2} - 9 n + 2 n + 3$

Simplify by adding terms.

Subtract $6 n^{2}$ from $3 n^{2}$ .

$2 n^{3} - 3 n^{2} - 9 n + 2 n + 3$

Add $- 9 n$ and $2 n$ .

$2 n^{3} - 3 n^{2} - 7 n + 3$