To determine the factors of the polynomial [tex]\( x^3 + 4x^2 + 5x + 20 \)[/tex] by grouping, we need to follow these steps:
1. Group the Terms: We begin by grouping the polynomial into two pairs of terms.
[tex]\[
(x^3 + 4x^2) + (5x + 20)
\][/tex]
2. Factor Each Group:
- For the first group [tex]\( x^3 + 4x^2 \)[/tex], we can factor out [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(x + 4)
\][/tex]
- For the second group [tex]\( 5x + 20 \)[/tex], we can factor out [tex]\( 5 \)[/tex]:
[tex]\[
5(x + 4)
\][/tex]
3. Combine Using the Common Factor:
- Notice that [tex]\( (x + 4) \)[/tex] is common in both factored groups:
[tex]\[
x^2(x + 4) + 5(x + 4)
\][/tex]
4. Factor Out the Common Factor:
- Since [tex]\( (x + 4) \)[/tex] is common, we factor it out:
[tex]\[
(x + 4)(x^2 + 5)
\][/tex]
So, the correct way to determine the factors of [tex]\( x^3 + 4x^2 + 5x + 20 \)[/tex] by grouping is shown in the option:
[tex]\[ x^2(x+4)+5(x+4) \][/tex]
This shows the step-by-step process of grouping and factoring the given polynomial.
