To solve the given equation:

\[
\frac{\frac{\sqrt{x}}{\sqrt{x} - 1} - \frac{1}{x - \sqrt{x}}}{\frac{\sqrt{x} + 1}{\sqrt{x} + 2}} = 2
\]

we need to simplify the expression step by step.

First, let's simplify the numerator of the main fraction:

\[
\frac{\sqrt{x}}{\sqrt{x} - 1} - \frac{1}{x - \sqrt{x}}
\]

To combine these two fractions, we need a common denominator. The common denominator for \(\sqrt{x} - 1\) and \(x - \sqrt{x}\) is \((\sqrt{x} - 1)(x - \sqrt{x})\).

Rewrite each fraction with the common denominator:

\[
\frac{\sqrt{x}(x - \sqrt{x})}{(\sqrt{x} - 1)(x - \sqrt{x})} - \frac{1(\sqrt{x} - 1)}{(x - \sqrt{x})(\sqrt{x} - 1)}
\]

Simplify the numerators:

\[
\frac{x\sqrt{x} - x}{(\sqrt{x} - 1)(x - \sqrt{x})} - \frac{\sqrt{x} - 1}{(\sqrt{x} - 1)(x - \sqrt{x})}
\]

Combine the fractions:

\[
\frac{x\sqrt{x} - x - (\sqrt{x} - 1)}{(\sqrt{x} - 1)(x - \sqrt{x})}
\]

Simplify the numerator:

\[
x\sqrt{x} - x - \sqrt{x} + 1
\]

So, the numerator of the main fraction is:

\[
\frac{x\sqrt{x} - x - \sqrt{x} + 1}{(\sqrt{x} - 1)(x - \sqrt{x})}
\]

Now, let's rewrite the entire expression:

\[
\frac{\frac{x\sqrt{x} - x - \sqrt{x} + 1}{(\sqrt{x} - 1)(x - \sqrt{x})}}{\frac{\sqrt{x} + 1}{\sqrt{x} + 2}} = 2
\]

This can be simplified by multiplying by the reciprocal of the denominator:

\[
\frac{x\sqrt{x} - x - \sqrt{x} + 1}{(\sqrt{x} - 1)(x - \sqrt{x})} \cdot \frac{\sqrt{x} + 2}{\sqrt{x} + 1} = 2
\]

Simplify the left-hand side:

\[
\frac{(x\sqrt{x} - x - \sqrt{x} + 1)(\sqrt{x} + 2)}{(\sqrt{x} - 1)(x - \sqrt{x})(\sqrt{x} + 1)} = 2
\]

Now, we need to solve for \(x\). Let's assume \(x = y^2\) where \(y = \sqrt{x}\):

\[
\frac{(y^3 - y^2 - y + 1)(y + 2)}{(y - 1)(y^2 - y)(y + 1)} = 2
\]

Simplify the expression:

\[
\frac{(y^4 + 2y^3 - y^3 - 2y^2 - y^2 - 2y + y + 2)}{(y - 1)(y^3 - y^2 + y^2 - y)} = 2
\]

Combine like terms:

\[
\frac{y^4 + y^3 - 3y^2 - y + 2}{(y - 1)(y^3 - y)} = 2
\]

Simplify the denominator:

\[
\frac{y^4 + y^3 - 3y^2 - y + 2}{y^4 - y^2 - y^3 + y} = 2
\]

Now, we can solve for \(y\) and then find \(x\). However, this equation is quite complex, and it might be easier to solve it numerically or graphically. Let's check if there are simpler values for \(y\) that satisfy the equation.

By inspection, if \(y = 2\):

\[
x = y^2 = 4
\]

Let's verify if \(x = 4\) satisfies the original equation:

\[
\frac{\frac{\sqrt{4}}{\sqrt{4} - 1} - \frac{1}{4 - \sqrt{4}}}{\frac{\sqrt{4} + 1}{\sqrt{4} + 2}} = 2
\]

\[
\frac{\frac{2}{2 - 1} - \frac{1}{4 - 2}}{\frac{2 + 1}{2 + 2}} = 2
\]

\[
\frac{2 - \frac{1}{2}}{\frac{3}{4}} = 2
\]

\[
\frac{2 - 0.5}{0.75} = 2
\]

\[
\frac{1.5}{0.75} = 2
\]

\[
2 = 2
\]

So, \(x = 4\) is indeed a solution. Therefore, the solution to the equation is:

\[
\boxed{4}
\]