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To multiply the expression \(2\sqrt{x} - \sqrt{x + h}\) by its conjugate, you use the technique of multiplying by the conjugate to simplify expressions involving square roots.

The conjugate of \(2\sqrt{x} - \sqrt{x + h}\) is \(2\sqrt{x} + \sqrt{x + h}\).

Here's how you multiply these two expressions:

1. **Identify the expression and its conjugate:**
   \[
   (2\sqrt{x} - \sqrt{x + h}) \text{ and } (2\sqrt{x} + \sqrt{x + h})
   \]

2. **Multiply the expression by its conjugate:**
   \[
   (2\sqrt{x} - \sqrt{x + h})(2\sqrt{x} + \sqrt{x + h})
   \]

3. **Use the difference of squares formula:** 
   The product of a binomial and its conjugate is the difference of squares:
   \[
   (a - b)(a + b) = a^2 - b^2
   \]

   Applying this to our terms:
   \[
   (2\sqrt{x})^2 - (\sqrt{x + h})^2
   \]

4. **Simplify each square:**
   \[
   (2\sqrt{x})^2 = 4x \quad \text{and} \quad (\sqrt{x + h})^2 = x + h
   \]

5. **Combine and simplify:**
   \[
   4x - (x + h) = 4x - x - h = 3x - h
   \]

Thus, the result of multiplying the expression by its conjugate is:

\[
(2\sqrt{x} - \sqrt{x + h})(2\sqrt{x} + \sqrt{x + h}) = 3x - h.
\]

Would you like further details on this process or have any specific questions?

### Related Questions:

1. What are conjugates, and why are they useful in simplifying square root expressions?
2. How would you simplify an expression involving the product of two conjugates with different terms?
3. What is the general formula for multiplying conjugates?
4. Can you explain how to rationalize a denominator using conjugates?
5. How do you handle multiplying conjugates when dealing with complex numbers?

**Tip:** Multiplying by conjugates is a powerful tool in algebra for removing radicals from denominators or simplifying expressions with roots. Always ensure to multiply the numerator and denominator by the same conjugate when rationalizing fractions.

Learn how to multiply the conjugates of expressions involving square roots step-by-step with this comprehensive guide. Understand the difference of squares formula and its application in simplifying algebraic expressions. Suitable for high school students studying algebra.

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