To apply the Distributive Property to the expression \((3x - 1)(x - 4)\), we can use the FOIL method (First, Outside, Inside, Last) which is a specific application of the Distributive Property.
1. **First:** \(3x \cdot x = 3x^2\)
2. **Outside:** \(3x \cdot (-4) = -12x\)
3. **Inside:** \(-1 \cdot x = -x\)
4. **Last:** \(-1 \cdot (-4) = 4\)
Now, combine these results:
\[
3x^2 - 12x - x + 4 = 3x^2 - 13x + 4
\]
However, upon reviewing the options given, we can simplify combinations instead. Let's analyze each option:
- **A. \(3x^2 + 6x + 4\)** - This does not match our result.
- **B. \(3x(- 1) + 3x(3x) + x(- 4) + x(x)\)** - This option is incorrect as it does not follow proper multiplication.
- **C. \(3x^2 - 5x + 4\)** - This does not match the results, though it's close.
- **D. \(3x(x) + 3x(- 4) + -1(x) + -1(- 4)\)** - This correctly expands the expression and results in a valid combination.
Thus, option **D** shows correct use of the Distributive Property.
Update:** If we observe what we've calculated \(3x^2 - 13x + 4\) actually, there might be an error confirming. The determination roots down to following checks.
None of the options presented appears to give us a matching output of what we derived correctly from the operations! However, we conclude D is the format of correct operation depending on flows, while it falls behind comparing simplistically to anything ranging to the final numerical values.
Our closest thought path evaluates back: original checks affirm. Define between errors occurring.
Final thought path to answer then validates into recognizing:
none of those leads directly to correctness.
The answer closest in journey thusly can be walked:
No distinct options. Thus align affirmatively transitioning thoughts of **D** intended signify the operation use-method.
So we minimally conclude reaffirming structure choice **D** resonates intention, if we clearly reaffirm recognize trailing variables and relevances in alignment toward workings yield clearer numerical outputs.
D stands to represent correct approach, and lead to emphasis hands on distributive reconciliation working through all numerical steps.