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To solve the quadratic equation \( 4x^2 - 8x - 21 = 0 \) by completing the square, let's go through the steps:

### Step 1: Divide the equation by the coefficient of \( x^2 \).
\[ 4x^2 - 8x - 21 = 0 \]
Divide everything by 4:
\[ x^2 - 2x - \frac{21}{4} = 0 \]

### Step 2: Move the constant term to the right side of the equation.
\[ x^2 - 2x = \frac{21}{4} \]

### Step 3: Complete the square.
To complete the square, take half the coefficient of \( x \), square it, and add it to both sides:
Half of \(-2\) is \(-1\), and squaring it gives \(1\):
\[ x^2 - 2x + 1 = \frac{21}{4} + 1 \]
\[ x^2 - 2x + 1 = \frac{21}{4} + \frac{4}{4} \]
\[ x^2 - 2x + 1 = \frac{25}{4} \]

### Step 4: Write the left side as a perfect square and simplify the right side.
\[ (x - 1)^2 = \frac{25}{4} \]

### Step 5: Solve for \( x \) by taking the square root of both sides.
\[ x - 1 = \pm \sqrt{\frac{25}{4}} \]
\[ x - 1 = \pm \frac{5}{2} \]

### Step 6: Solve for \( x \).
\[ x = 1 \pm \frac{5}{2} \]

This gives us two solutions:
1. \( x = 1 + \frac{5}{2} = \frac{7}{2} \)
2. \( x = 1 - \frac{5}{2} = -\frac{3}{2} \)

### Check the given options:
None of the options given directly match the solutions we've found. However, let's revisit the equation to check if there might have been a different interpretation:

Given the complex numbers in the options, let's verify if a potential miscalculation or different format could be intended.

Instead of reviewing possible issues or non-standard solutions, let's compare directly:
- **None of the options A-F match our derived solutions exactly** using the standard form. Given complex numbers like \(1 + 2i\) might appear in alternative formats. But direct interpretation does not align.

### Conclusion:
None of the provided answers directly corresponds with the typical complete square solution. A different approach may be suggested or an alternative complex interpretation. 

Do you have any further questions or need additional verification on the provided answers? 

### 5 Related Questions:
1. What are the steps to complete the square for equations with complex solutions?
2. How does completing the square compare to the quadratic formula for solving quadratic equations?
3. How can complex solutions emerge from completing the square, and what does that indicate?
4. Can completing the square be used to derive the quadratic formula?
5. How do transformations of a quadratic function affect its solutions?

### Tip:
When completing the square, remember to always maintain balance across the equation while adding and subtracting values—especially when dealing with fractions or coefficients greater than 1.

Use the completing the square process to identify solutions to the quadratic equation 4x^2 - 8x - 21 = 0.

This detailed explanation covers how to solve the quadratic equation 4x^2 - 8x - 21 = 0 by completing the square. The problem involves key algebraic concepts like completing the square and the quadratic formula, suitable for students in Grades 10-12.

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