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To solve the quadratic equation \(x^2 + 2x = 25\) by completing the square, we follow these steps:

1. Move the constant term to the right side:
   \[
   x^2 + 2x = 25
   \]

2. Add and subtract the square of half the coefficient of \(x\) to the left side. The coefficient of \(x\) is 2, so half of it is 1, and its square is 1. We add and subtract 1 inside the equation:
   \[
   x^2 + 2x + 1 - 1 = 25
   \]

3. Rewrite the equation by grouping the perfect square trinomial and moving the subtracted term to the right side:
   \[
   (x + 1)^2 - 1 = 25
   \]
   \[
   (x + 1)^2 = 25 + 1
   \]
   \[
   (x + 1)^2 = 26
   \]

4. Solve for \(x\) by taking the square root of both sides:
   \[
   x + 1 = \pm \sqrt{26}
   \]

5. Isolate \(x\) to find the solutions:
   \[
   x = -1 \pm \sqrt{26}
   \]

The solutions to the quadratic equation are:
\[
x = -1 \pm \sqrt{26}
\]

This matches option D. So, the answer is:

\[
\boxed{\text{D. } x = -1 \pm \sqrt{26}}
\]

Do you want details or have any questions? Here are five more questions you might ask:

1. How do you complete the square for a quadratic equation?
2. What are the steps to derive the quadratic formula from completing the square?
3. How do you solve a quadratic equation using the quadratic formula?
4. What is the significance of the discriminant in determining the nature of the roots?
5. How can you verify the solutions of a quadratic equation?

**Tip:** Completing the square is a useful technique for solving quadratic equations, particularly when the quadratic formula is not immediately obvious or for deriving the vertex form of the equation.

Learn how to solve the quadratic equation x^2 + 2x = 25 by completing the square with a step-by-step explanation. This problem involves key algebraic concepts such as completing the square, finding roots of equations, and using the quadratic formula. Suitable for students in Grades 10-12.

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