The Distributive Property Quick Check

To rewrite the expression \((3x-1)(x-4)\) using the Distributive Property, we need to distribute each term in the first parentheses across each term in the second parentheses.

Let's do the distribution step by step:

1. Distribute \(3x\):
\[
3x(x) + 3x(-4) = 3x^2 - 12x
\]

2. Distribute \(-1\):
\[
-1(x) + -1(-4) = -x + 4
\]

3. Combine the results:
\[
3x^2 - 12x - x + 4 = 3x^2 - 13x + 4
\]

Now let’s compare this result to the options given:
- \(3x^{2}-5x+4\)
- \(3x(-1)+3x(3x)+x(-4)+x(x)\)
- \(3x(x)+3x(-4)+-1(x)+-1(-4)\)
- \(3x^{2}+6x+4\)

None of the options seem to reflect the correct expanded result, which is \(3x^2 - 13x + 4\).

The correct answer based on the question's requirement for the expression would be: **None of the listed options** accurately represent the correct expansion of \((3x-1)(x-4)\). However, the closest structured step among the choices is:

\[
3x(x)+3x(-4)+-1(x)+-1(-4)
\]

This explicitly breaks down the distribution process, even if it does not yield the final simplified form.