To multiply the polynomials [tex]\((x-2)\)[/tex] and [tex]\((3x^2 + 4x - 3)\)[/tex], we will use the distributive property (also known as the FOIL method for binomials). Here’s the step-by-step solution:
1. First, distribute [tex]\(x\)[/tex] over the second polynomial:
[tex]\[
x \cdot (3x^2 + 4x - 3) = x \cdot 3x^2 + x \cdot 4x + x \cdot (-3)
\][/tex]
Simplifying each term:
[tex]\[
x \cdot 3x^2 = 3x^3
\][/tex]
[tex]\[
x \cdot 4x = 4x^2
\][/tex]
[tex]\[
x \cdot (-3) = -3x
\][/tex]
Therefore:
[tex]\[
x \cdot (3x^2 + 4x - 3) = 3x^3 + 4x^2 - 3x
\][/tex]
2. Next, distribute [tex]\(-2\)[/tex] over the second polynomial:
[tex]\[
-2 \cdot (3x^2 + 4x - 3) = -2 \cdot 3x^2 + -2 \cdot 4x + -2 \cdot (-3)
\][/tex]
Simplifying each term:
[tex]\[
-2 \cdot 3x^2 = -6x^2
\][/tex]
[tex]\[
-2 \cdot 4x = -8x
\][/tex]
[tex]\[
-2 \cdot (-3) = 6
\][/tex]
Therefore:
[tex]\[
-2 \cdot (3x^2 + 4x - 3) = -6x^2 - 8x + 6
\][/tex]
3. Add the results of the two distributions:
[tex]\[
(3x^3 + 4x^2 - 3x) + (-6x^2 - 8x + 6)
\][/tex]
Combine like terms:
[tex]\[
3x^3 + 4x^2 - 6x^2 - 3x - 8x + 6
\][/tex]
[tex]\[
3x^3 + (4x^2 - 6x^2) + (-3x - 8x) + 6
\][/tex]
[tex]\[
3x^3 - 2x^2 - 11x + 6
\][/tex]
Thus, the expanded form of [tex]\((x-2)(3x^2 + 4x - 3)\)[/tex] is [tex]\(\boxed{3x^3 - 2x^2 - 11x + 6}\)[/tex].
Comparing this with the given options, the correct answer is:
[tex]\[
3x^3 - 2x^2 - 11x + 6
\][/tex]
This matches the third option. Therefore, the correct option is:
[tex]\[
\boxed{3}
\][/tex]
