To multiply [tex]\((x-4)(x^2 + 6x - 5)\)[/tex], we will use the distributive property, also known as the FOIL method for polynomials. This involves distributing each term in the first polynomial by each term in the second polynomial.
First, we'll distribute the [tex]\(x\)[/tex] term in [tex]\((x-4)\)[/tex]:
1. [tex]\(x \cdot x^2 = x^3\)[/tex]
2. [tex]\(x \cdot 6x = 6x^2\)[/tex]
3. [tex]\(x \cdot (-5) = -5x\)[/tex]
Now, we distribute the [tex]\(-4\)[/tex] term in [tex]\((x-4)\)[/tex]:
1. [tex]\(-4 \cdot x^2 = -4x^2\)[/tex]
2. [tex]\(-4 \cdot 6x = -24x\)[/tex]
3. [tex]\(-4 \cdot (-5) = 20\)[/tex]
Next, we combine all these terms together:
[tex]\[ x^3 + 6x^2 - 5x - 4x^2 - 24x + 20 \][/tex]
Now, we need to combine like terms:
[tex]\[ x^3 + (6x^2 - 4x^2) + (-5x - 24x) + 20 \][/tex]
[tex]\[ x^3 + 2x^2 - 29x + 20 \][/tex]
Therefore, the expanded expression is:
[tex]\[ x^3 + 2x^2 - 29x + 20 \][/tex]
Comparing this with the given options, we find that the correct option is:
[tex]\[ x^3 + 2x^2 - 29x + 20 \][/tex]
Thus, the correct answer corresponds to option 1:
[tex]\[ \boxed{1} \][/tex]
