Let's analyze Faelyn's work step by step.
Faelyn's original polynomial is:
[tex]\[6 x^4 - 8 x^2 + 3 x^2 + 4\][/tex]
Step 1:
Faelyn grouped the terms as:
[tex]\[\left(6 x^4 - 8 x^2\right) + \left(3 x^2 + 4\right)\][/tex]
Step 2:
She factored the GCF out of the groups:
[tex]\[2 x^2 (3 x^2 - 4) + 1 (3 x^2 + 4)\][/tex]
Here Faelyn noticed she doesn't have a common factor between the groups.
Let's identify where Faelyn should go next.
Upon closer inspection, Faelyn should go back and regroup the terms in Step 1:
[tex]\[\left(6 x^4 + 3 x^2\right) - \left(8 x^2 + 4\right)\][/tex]
Next, factor the GCF out of each group:
Regrouped:
[tex]\[\left(6 x^4 + 3 x^2\right) - \left(8 x^2 + 4\right)\][/tex]
Factor out the greatest common factors from each group:
[tex]\[3 x^2 \left(2 x^2 + 1\right) - 4 \left(2 x^2 + 1\right)\][/tex]
Now, we see that [tex]\((2 x^2 + 1)\)[/tex] is a common factor. So, we can factor [tex]\((2 x^2 + 1)\)[/tex] out:
[tex]\[\left(2 x^2 + 1\right)\left(3 x^2 - 4\right)\][/tex]
This indicates what Faelyn should have done:
The accurate action for Faelyn:
Faelyn should go back and regroup the terms in Step 1 as [tex]\(\left(6 x^4 + 3 x^2\right) - \left(8 x^2 + 4\right)\)[/tex].
Therefore, the correct choice is:
Faelyn should go back and regroup the terms in Step 1 as [tex]\(\left(6 x^4 + 3 x^2\right) - \left(8 x^2 + 4\right)\)[/tex].
