Let’s carefully look through each step of Talia's work and figure out what went wrong and what she should do next.
### Given Polynomial
[tex]\[ 15x^2 - 3x - 20x + 4 \][/tex]
### Step 1: Grouping the Terms
[tex]\[ (15x^2 - 3x) + (-20x + 4) \][/tex]
### Step 2: Factoring Out the GCF
From the first group [tex]\((15x^2 - 3x)\)[/tex], factor out the GCF, which is [tex]\(3x\)[/tex]:
[tex]\[ 3x(5x - 1) \][/tex]
From the second group [tex]\((-20x + 4)\)[/tex], factor out the GCF, which is [tex]\(4\)[/tex]. However, Talia did:
[tex]\[ 4(-5x + 1) \][/tex]
Here, Talia noticed that the binomial factors [tex]\((5x - 1)\)[/tex] and [tex]\((-5x + 1)\)[/tex] are not the same. This discrepancy suggests that another step is required to factor the polynomial correctly.
### Corrective Step
Talia’s insight is correct; she should factor out a negative from one of the groups to make the binomials identical. Specifically, factor out [tex]\(-4\)[/tex] instead of [tex]\(4\)[/tex] from the second group:
[tex]\[ -4(5x - 1) \][/tex]
Now, rewrite the polynomial grouping and factor out the common binomial factor [tex]\((5x - 1)\)[/tex]:
### Adjusted Factoring
[tex]\[ 15x^2 - 3x - 20x + 4 \][/tex]
[tex]\[ = (15x^2 - 3x) + (-20x + 4) \][/tex]
[tex]\[ = 3x(5x - 1) - 4(5x - 1) \][/tex]
### Factor the Common Binomial
Since [tex]\((5x - 1)\)[/tex] is now a common factor, factor it out:
[tex]\[ = (3x - 4)(5x - 1) \][/tex]
Thus, Talia should factor out a negative from one of the groups so the binomials will be the same.
So the correct option is:
[tex]\[ \boxed{Talia\ needs\ to\ factor\ out\ a\ negative\ from\ one\ of\ the\ groups\ so\ the\ binomials\ will\ be\ the\ same.} \][/tex]
