Simplify (2x square root of x^2+1)^2

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

Use the power rule ${(a b)}^{n} = a^{n} b^{n}$ to distribute the exponent.

Apply the product rule to $2 x \sqrt{x^{2} + 1}$ .

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

Apply the product rule to $2 x$ .

$2^{2} x^{2} {\sqrt{x^{2} + 1}}^{2}$

Raise $2$ to the power of $2$ .

$4 x^{2} {\sqrt{x^{2} + 1}}^{2}$

Rewrite ${\sqrt{x^{2} + 1}}^{2}$ as $x^{2} + 1$ .

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{x^{2} + 1}$ as ${(x^{2} + 1)}^{\frac{1}{2}}$ .

$4 x^{2} {({(x^{2} + 1)}^{\frac{1}{2}})}^{2}$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Combine $\frac{1}{2}$ and $2$ .

$4 x^{2} {(x^{2} + 1)}^{\frac{2}{2}}$

Cancel the common factor of $2$ .

Cancel the common factor.

$4 x^{2} {(x^{2} + 1)}^{\frac{2}{2}}$

Rewrite the expression.

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

Simplify.

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

Apply the distributive property.

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

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

Move $x^{2}$ .

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

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

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

Add $2$ and $2$ .

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

Multiply $4$ by $1$ .

$4 x^{4} + 4 x^{2}$