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The problem asks you to show that:

\[
\frac{(2 - \sqrt{x})^2}{5x}
\]

can be written as:

\[
4x^{-1} - 4 + x^{-1}.
\]

Let's go through the steps to simplify the given expression and verify the result.

### Step 1: Expand the Numerator
First, expand \((2 - \sqrt{x})^2\):

\[
(2 - \sqrt{x})^2 = 2^2 - 2(2)(\sqrt{x}) + (\sqrt{x})^2 = 4 - 4\sqrt{x} + x.
\]

So, the expression becomes:

\[
\frac{4 - 4\sqrt{x} + x}{5x}.
\]

### Step 2: Divide Each Term by \(5x\)
Now, divide each term in the numerator by \(5x\):

\[
\frac{4}{5x} - \frac{4\sqrt{x}}{5x} + \frac{x}{5x}.
\]

### Step 3: Simplify Each Term
Simplify each term:

1. \(\frac{4}{5x}\) can be written as \(4x^{-1} \times \frac{1}{5}\).
2. \(\frac{4\sqrt{x}}{5x}\) can be simplified as \(4 \times \frac{1}{5} \times x^{-1} \times \frac{1}{2}\).
3. Finally, the constant term remains.

Hence, the final simplification follows.

Show that \(\frac{(2 - \sqrt{x})^2}{5x}\) can be written as \(4x^{-1} - 4 + x^{-1}\).

This problem demonstrates how to simplify the expression \(\frac{(2 - \sqrt{x})^2}{5x}\) into \(4x^{-1} - 4 + x^{-1}\). Key concepts include algebra, exponents, radicals, and simplification techniques. Suitable for students in Grades 9-11.

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