Let's go through the steps in detail to address what Faelyn should do next.
1. Original Grouping:
Given polynomial: [tex]\(6x^4 - 8x^2 + 3x^2 + 4\)[/tex]
Grouping the terms:
[tex]\[
(6x^4 - 8x^2) + (3x^2 + 4)
\][/tex]
2. Factoring Out the GCF in Each Group:
Factor out the greatest common factor (GCF) from each grouped term:
[tex]\[
2x^2(3x^2 - 4) + 1(3x^2 + 4)
\][/tex]
At this stage, Faelyn sees that there is no common binomial factor between the two grouped expressions, i.e., [tex]\(3x^2 - 4\)[/tex] and [tex]\(3x^2 + 4\)[/tex].
3. Regrouping the Terms:
To make the binomial factors the same, Faelyn should factor out a negative sign from one of the groups. This will align the binomials properly:
[tex]\[
(6x^4 - 8x^2) - (-(3x^2 + 4))
\][/tex]
4. Factoring Out the GCF Again:
By factoring a negative from the second group, we can now express it as:
[tex]\[
6x^4 - 8x^2 - (3x^2 + 4)
\][/tex]
Simplifying the second group:
[tex]\[
(6x^4 - 8x^2) - (-3x^2 - 4)
\][/tex]
Now factor out the common binomial factor from the expression:
[tex]\[
2x^2(3x^2 - 4) - 1(3x^2 - 4)
\][/tex]
5. Combining the Factored Terms:
With the same binomial factor [tex]\(3x^2 - 4\)[/tex] in place, we can combine the factored terms:
[tex]\[
(3x^2 - 4)(2x^2 - 1)
\][/tex]
Thus, the polynomial [tex]\(6x^4 - 8x^2 + 3x^2 + 4\)[/tex] can be factored as [tex]\((3x^2 - 4)(2x^2 - 1)\)[/tex].
Hence, the accurate description of what Faelyn should do next is:
Faelyn should factor out a negative from one of the groups so the binomials will be the same.
