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To express \((2x - 5)(x + 4)\) in the form \(2(x + a)^2 - b\), we need to expand the expression, simplify it, and then rewrite it in the specified form.

### Step 1: Expand \((2x - 5)(x + 4)\)
\[
(2x - 5)(x + 4) = 2x \cdot x + 2x \cdot 4 - 5 \cdot x - 5 \cdot 4
\]
\[
= 2x^2 + 8x - 5x - 20
\]
\[
= 2x^2 + 3x - 20
\]

### Step 2: Rewrite \(2x^2 + 3x - 20\) in the form \(2(x + a)^2 - b\)

1. **Factor out 2 from the terms involving \(x\):**
   \[
   2x^2 + 3x - 20 = 2(x^2 + \frac{3}{2}x) - 20
   \]

2. **Complete the square for \(x^2 + \frac{3}{2}x\):**
   - Take half of \(\frac{3}{2}\), square it, and add inside the parentheses.
   - Half of \(\frac{3}{2}\) is \(\frac{3}{4}\), and squaring it gives \(\frac{9}{16}\).

   So, we rewrite:
   \[
   2\left(x^2 + \frac{3}{2}x + \frac{9}{16} - \frac{9}{16}\right) - 20
   \]
   \[
   = 2\left((x + \frac{3}{4})^2 - \frac{9}{16}\right) - 20
   \]

3. **Distribute the 2 and simplify:**
   \[
   = 2(x + \frac{3}{4})^2 - 2 \cdot \frac{9}{16} - 20
   \]
   \[
   = 2(x + \frac{3}{4})^2 - \frac{9}{8} - 20
   \]
   \[
   = 2(x + \frac{3}{4})^2 - \frac{169}{8}
   \]

### Final Answer
\[
2(x + \frac{3}{4})^2 - \frac{169}{8}
\]

This is in the form \(2(x + a)^2 - b\), where \(a = \frac{3}{4}\) and \(b = \frac{169}{8}\).

Would you like a detailed explanation of any step, or do you have any questions?

Here are five related questions to deepen your understanding:
1. How do you complete the square for any quadratic expression?
2. Why do we factor out 2 before completing the square in this problem?
3. How does expanding \((x + a)^2\) help in understanding the transformation?
4. What are the applications of expressing quadratic expressions in completed square form?
5. How do you verify the final answer by expanding back?

**Tip:** Completing the square is a powerful technique to transform quadratic expressions and is especially useful for solving equations and analyzing parabolas.

Write (2x - 5)(x + 4) in the form 2(x + a)^2 - b.

Learn how to rewrite the expression (2x - 5)(x + 4) in the form 2(x + a)^2 - b by expanding and completing the square. This solution covers essential algebraic techniques suitable for students in Grades 10-12.

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