To determine the factors of the polynomial [tex]\(x^3 + 4x^2 + 5x + 20\)[/tex] by grouping, we will follow these steps:
1. Group the terms into pairs:
[tex]\[
(x^3 + 4x^2) + (5x + 20)
\][/tex]
2. Factor out the common factor from each pair:
From the first group [tex]\((x^3 + 4x^2)\)[/tex], the common factor is [tex]\(x^2\)[/tex]:
[tex]\[
x^2(x + 4)
\][/tex]
From the second group [tex]\((5x + 20)\)[/tex], the common factor is 5:
[tex]\[
5(x + 4)
\][/tex]
3. Notice the common binomial factor [tex]\((x + 4)\)[/tex]:
We can now factor out the common binomial [tex]\((x + 4)\)[/tex]:
[tex]\[
x^2(x + 4) + 5(x + 4)
\][/tex]
Here, we see that both terms contain a common binomial factor [tex]\((x + 4)\)[/tex].
4. Factor out the common binomial factor:
[tex]\[
(x^2 + 5)(x + 4)
\][/tex]
Therefore, one way to determine the factors of [tex]\(x^3 + 4x^2 + 5x + 20\)[/tex] by grouping shows that the correct step-by-step solution for factoring by grouping is:
[tex]\[
x^2(x + 5) + 4(x + 5)
\][/tex]
This correctly factors the polynomial as indicated by the grouping.
