To factor the polynomial [tex]\(10x^3 + 3x^2 - 20x - 6\)[/tex] by grouping, follow these steps carefully:
1. Look at the polynomial and split it into two groups:
[tex]\[
\left(10x^3 + 3x^2\right) + \left(-20x - 6\right)
\][/tex]
2. Factor out the common factor from each group:
- In the first group [tex]\(\left(10x^3 + 3x^2\right)\)[/tex], the common factor is [tex]\(x^2\)[/tex]:
[tex]\[
10x^3 + 3x^2 = x^2(10x + 3)
\][/tex]
- In the second group [tex]\(\left(-20x - 6\right)\)[/tex], the common factor is [tex]\(-2\)[/tex]:
[tex]\[
-20x - 6 = -2(10x + 3)
\][/tex]
3. Write the polynomial grouping with the common factors factored out:
[tex]\[
x^2(10x + 3) - 2(10x + 3)
\][/tex]
Given the choices provided:
1. [tex]\(x^2\)[/tex] and [tex]\(-2x\)[/tex]
2. [tex]\(2x^2\)[/tex] and [tex]\(-2x\)[/tex]
3. [tex]\(x^2\)[/tex] and [tex]\(-2\)[/tex]
4. [tex]\(2x^2\)[/tex] and [tex]\(-2\)[/tex]
The correct common factors to use in the next step of factoring are:
[tex]\[
x^2 \text{ and } -2
\][/tex]
