To simplify the expression [tex]\((2g - 1)(3g^2 - 2g + 4)\)[/tex], we need to use the distributive property, also known as the FOIL method for multiplying binomials, to expand the product of these polynomials. Let’s go through the steps in detail:
1. Distribute each term in the first polynomial to each term in the second polynomial:
[tex]\[
(2g - 1)(3g^2 - 2g + 4)
\][/tex]
This can be broken down into:
[tex]\[
2g \cdot (3g^2 - 2g + 4) - 1 \cdot (3g^2 - 2g + 4)
\][/tex]
2. Multiply [tex]\(2g\)[/tex] by each term in the second polynomial:
[tex]\[
2g \cdot 3g^2 = 6g^3
\][/tex]
[tex]\[
2g \cdot (-2g) = -4g^2
\][/tex]
[tex]\[
2g \cdot 4 = 8g
\][/tex]
So, distributing [tex]\(2g\)[/tex] gives us:
[tex]\[
6g^3 - 4g^2 + 8g
\][/tex]
3. Multiply [tex]\(-1\)[/tex] by each term in the second polynomial:
[tex]\[
-1 \cdot 3g^2 = -3g^2
\][/tex]
[tex]\[
-1 \cdot (-2g) = 2g
\][/tex]
[tex]\[
-1 \cdot 4 = -4
\][/tex]
So, distributing [tex]\(-1\)[/tex] gives us:
[tex]\[
-3g^2 + 2g - 4
\][/tex]
4. Combine the results from steps 2 and 3:
Adding the results together, we have:
[tex]\[
(6g^3 - 4g^2 + 8g) + (-3g^2 + 2g - 4)
\][/tex]
5. Combine like terms:
Group the like terms:
[tex]\[
6g^3 + (-4g^2 - 3g^2) + (8g + 2g) - 4
\][/tex]
Simplify each group:
[tex]\[
6g^3 - 7g^2 + 10g - 4
\][/tex]
Therefore, the simplified form of the expression [tex]\((2g - 1)(3g^2 - 2g + 4)\)[/tex] is:
[tex]\[
6g^3 - 7g^2 + 10g - 4
\][/tex]
Upon comparing this with the given options:
- [tex]\(6g^3 + g^2 + 6g - 4\)[/tex]
- [tex]\(6g^3 - 3g^2 + 8g - 4\)[/tex]
- [tex]\(6g^3 + g^2 + 10g - 4\)[/tex]
- [tex]\(6g^3 - 7g^2 + 10g - 4\)[/tex]
The correct simplified expression is:
[tex]\(\boxed{6g^3 - 7g^2 + 10g - 4}\)[/tex]
