Find the Product 3x^5(2x^2+4x+1)

$3 x^{5} (2 x^{2} + 4 x + 1)$

Apply the distributive property.

$3 x^{5} (2 x^{2}) + 3 x^{5} (4 x) + 3 x^{5} \cdot 1$

Simplify.

Rewrite using the commutative property of multiplication.

$3 \cdot 2 x^{5} x^{2} + 3 x^{5} (4 x) + 3 x^{5} \cdot 1$

Rewrite using the commutative property of multiplication.

$3 \cdot 2 x^{5} x^{2} + 3 \cdot 4 x^{5} x + 3 x^{5} \cdot 1$

Multiply $3$ by $1$ .

$3 \cdot 2 x^{5} x^{2} + 3 \cdot 4 x^{5} x + 3 x^{5}$

Simplify each term.

Multiply $x^{5}$ by $x^{2}$ by adding the exponents.

Move $x^{2}$ .

$3 \cdot 2 (x^{2} x^{5}) + 3 \cdot 4 x^{5} x + 3 x^{5}$

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

$3 \cdot 2 x^{2 + 5} + 3 \cdot 4 x^{5} x + 3 x^{5}$

Add $2$ and $5$ .

$3 \cdot 2 x^{7} + 3 \cdot 4 x^{5} x + 3 x^{5}$

Multiply $3$ by $2$ .

$6 x^{7} + 3 \cdot 4 x^{5} x + 3 x^{5}$

Multiply $x^{5}$ by $x$ by adding the exponents.

Move $x$ .

$6 x^{7} + 3 \cdot 4 (x \cdot x^{5}) + 3 x^{5}$

Multiply $x$ by $x^{5}$ .

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

$6 x^{7} + 3 \cdot 4 (x^{1} x^{5}) + 3 x^{5}$

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

$6 x^{7} + 3 \cdot 4 x^{1 + 5} + 3 x^{5}$

Add $1$ and $5$ .

$6 x^{7} + 3 \cdot 4 x^{6} + 3 x^{5}$

Multiply $3$ by $4$ .

$6 x^{7} + 12 x^{6} + 3 x^{5}$