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------------------------------------------------ Factor by grouping. Be sure to factor this polynomial completely:

[tex]2x^3 + 5x^2 - 18x - 45[/tex]

Sure! Let's factor the polynomial [tex]$2x^3 + 5x^2 - 18x - 45$[/tex] by grouping, step by step.

1. Group the terms:
- The polynomial [tex]$2x^3 + 5x^2 - 18x - 45$[/tex] can be divided into two groups: [tex]$(2x^3 + 5x^2)$[/tex] and [tex]$(-18x - 45)$[/tex].

2. Factor out the greatest common factor in each group:
- In the first group [tex]$(2x^3 + 5x^2)$[/tex], the greatest common factor is [tex]$x^2$[/tex]. So, we factor it out:
[tex]\[
x^2(2x + 5)
\][/tex]
- In the second group [tex]$(-18x - 45)$[/tex], the greatest common factor is [tex]$-9$[/tex]. So, we factor it out:
[tex]\[
-9(2x + 5)
\][/tex]

3. Recognize and combine the common factor:
- We notice that [tex]$(2x + 5)$[/tex] is a common factor in both groups. So, we can combine the groups:
[tex]\[
(x^2 - 9)(2x + 5)
\][/tex]

4. Factor completely if possible:
- The expression [tex]$(x^2 - 9)$[/tex] is a difference of squares, which can be factored further:
[tex]\[
x^2 - 9 = (x - 3)(x + 3)
\][/tex]

5. Final factored form:
- Substitute [tex]$(x - 3)(x + 3)$[/tex] back into the expression:
[tex]\[
(x - 3)(x + 3)(2x + 5)
\][/tex]

Thus, the completely factored form of the polynomial [tex]$2x^3 + 5x^2 - 18x - 45$[/tex] is [tex]$(x - 3)(x + 3)(2x + 5)$[/tex].