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The problem asks to find the value of \( k \), given that \( (x - 2) \) is a factor of the quadratic expression \( 2x^2 + 12kx - 25k \).

### Step-by-Step Solution:

1. **Factor Theorem**:
   According to the factor theorem, if \( (x - 2) \) is a factor of \( 2x^2 + 12kx - 25k \), then the value of the expression at \( x = 2 \) must be zero:
   \[
   2(2)^2 + 12k(2) - 25k = 0
   \]

2. **Substitute \( x = 2 \) into the equation**:
   \[
   2(4) + 12k(2) - 25k = 0
   \]
   Simplifying this:
   \[
   8 + 24k - 25k = 0
   \]

3. **Simplify the equation**:
   \[
   8 - k = 0
   \]
   \[
   k = 8
   \]

Thus, the value of \( k \) is **8**.

The answer is correct: \( k = 8 \).

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### Related Questions:
1. How does the factor theorem help in determining the value of \( k \)?
2. Can the factorization of the quadratic expression \( 2x^2 + 12kx - 25k \) be verified using synthetic division?
3. What other methods could be used to solve for \( k \)?
4. How would the solution change if the factor given was \( (x - 3) \) instead of \( (x - 2) \)?
5. What would happen if \( (x + 2) \) was given as a factor instead?

**Tip**: When using the factor theorem, always remember to substitute the root of the factor into the polynomial to verify or find unknowns.

If (x - 2) is a factor of (2x^2 + 12kx - 25k), then what is the value of k?

Learn how to find the value of k by applying the factor theorem to the polynomial expression 2x^2 + 12kx - 25k. This solution demonstrates how to substitute and simplify, perfect for students in Grades 9-11.

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