To factor the polynomial [tex]\( x^3 + 4x^2 - 9x - 36 \)[/tex] by grouping, you follow these steps:
### Step 1: Group the terms in pairs
Consider the polynomial [tex]\( x^3 + 4x^2 - 9x - 36 \)[/tex]. We can group the terms to see if we can factor by grouping:
[tex]\[
x^3 + 4x^2 - 9x - 36 = (x^3 + 4x^2) + (-9x - 36)
\][/tex]
### Step 2: Factor out the greatest common factor (GCF) from each group
- From the first group [tex]\( (x^3 + 4x^2) \)[/tex], factor out [tex]\( x^2 \)[/tex]:
[tex]\[
x^2(x + 4)
\][/tex]
- From the second group [tex]\( (-9x - 36) \)[/tex], factor out [tex]\(-9\)[/tex]:
[tex]\[
-9(x + 4)
\][/tex]
### Step 3: Factor out the common binomial factor
Now, our expression looks like:
[tex]\[
x^2(x + 4) - 9(x + 4)
\][/tex]
Since [tex]\( (x + 4) \)[/tex] is common in both terms, factor [tex]\( (x + 4) \)[/tex] out:
[tex]\[
(x + 4)(x^2 - 9)
\][/tex]
### Step 4: Further factor any remaining expressions, if possible
The term [tex]\( x^2 - 9 \)[/tex] is a difference of squares and can be factored further:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]
However, for the purpose of matching it to the provided options, we don't need to further factor but should directly compare it to the options given:
- [tex]\((x^2 + 9)(x + 4)\)[/tex]
- [tex]\((x^2 - 9)(x - 4)\)[/tex]
- [tex]\((x^2 + 9)(x - 4)\)[/tex]
- [tex]\((x^2 - 9)(x + 4)\)[/tex]
We see that the expression [tex]\( (x + 4)(x^2 - 9) \)[/tex] matches [tex]\((x^2 - 9)(x + 4)\)[/tex].
### Conclusion
Therefore, the correct factorization of the polynomial [tex]\( x^3 + 4x^2 - 9x - 36 \)[/tex] is [tex]\((x^2 - 9)(x + 4)\)[/tex], which corresponds to option:
[tex]\[
\boxed{4}
\][/tex]
