To rewrite the expression \((3x - 1)(x - 4)\) using the Distributive Property (also known as the FOIL method for two binomials), we can calculate it as follows:
- First: Multiply the first terms: \(3x \cdot x = 3x^2\)
- Outside: Multiply the outside terms: \(3x \cdot (-4) = -12x\)
- Inside: Multiply the inside terms: \(-1 \cdot x = -1x\)
- Last: Multiply the last terms: \(-1 \cdot (-4) = 4\)
Now combine all these results:
\[ 3x^2 - 12x - 1x + 4 = 3x^2 - 13x + 4 \]
None of the provided answers match this exact result, but the closest structure found in the options directly corresponds to the algebraic manipulation using the Distributive Property.
If we analyze the options provided:
- A: \(3x^2 + 6x + 4\) - Not correct.
- B: \(3x(-1) + 3x(3x) + x(-4) + x(x)\) - Not correct.
- C: \(3x^2 - 5x + 4\) - Not correct.
- D: \(3x(x) + 3x(-4) + -1(x) + -1(-4)\) - This can be simplified and shows that calculations along the correct lines.
Answer: D seems to be a representation factoring relevant components, but still does not fully arrive at the expanded form. Thus, while none appear correct outright, D gives a clearer breakdown even without correcting for full numeric representation.
But for the exact value of the expression \((3x-1)(x-4)\) the accurate answer is rather \(3x^2 - 13x + 4\).