Simplify x(x+1)(2x+1)

$x (x + 1) (2 x + 1)$

Simplify by multiplying through.

Apply the distributive property.

$(x \cdot x + x \cdot 1) (2 x + 1)$

Simplify the expression.

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

$(x^{2} + x) (2 x + 1)$

Expand $(x^{2} + x) (2 x + 1)$ using the FOIL Method.

Apply the distributive property.

$x^{2} (2 x + 1) + x (2 x + 1)$

Apply the distributive property.

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

Apply the distributive property.

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

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

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

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

Move $x$ .

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

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

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

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

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

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

Add $1$ and $2$ .

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

Multiply $x^{2}$ by $1$ .

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

Rewrite using the commutative property of multiplication.

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

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

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

Multiply $x$ by $x$ .

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

Multiply $x$ by $1$ .

$2 x^{3} + x^{2} + 2 x^{2} + x$

Add $x^{2}$ and $2 x^{2}$ .

$2 x^{3} + 3 x^{2} + x$