Answer :

Sure, let's factor the polynomial [tex]\(8x^3 + 20x^2 - 18x - 45\)[/tex] completely.

### Step 1: Grouping
First, we try to group the terms to simplify the process:
- Group the first two terms: [tex]\(8x^3 + 20x^2\)[/tex]
- Group the last two terms: [tex]\(-18x - 45\)[/tex]

### Step 2: Factor out the greatest common factor (GCF) from each group
- In the first group [tex]\(8x^3 + 20x^2\)[/tex], the GCF is [tex]\(4x^2\)[/tex]. Factoring that out, we get:
[tex]\[4x^2(2x + 5)\][/tex]
- In the second group [tex]\(-18x - 45\)[/tex], the GCF is [tex]\(-9\)[/tex]. Factoring that out gives:
[tex]\[-9(2x + 5)\][/tex]

Now, we have:
[tex]\[4x^2(2x + 5) - 9(2x + 5)\][/tex]

### Step 3: Factor by grouping
Notice that [tex]\(2x + 5\)[/tex] is a common factor in both groups:
[tex]\[(4x^2 - 9)(2x + 5)\][/tex]

### Step 4: Factor the difference of squares
The expression [tex]\(4x^2 - 9\)[/tex] is a difference of squares, which can be factored further:
[tex]\[4x^2 - 9 = (2x - 3)(2x + 3)\][/tex]

### Step 5: Write the complete factorization
Substituting back, the fully factored expression is:
[tex]\[(2x - 3)(2x + 3)(2x + 5)\][/tex]

So, the complete factorization of the polynomial [tex]\(8x^3 + 20x^2 - 18x - 45\)[/tex] is [tex]\((2x - 3)(2x + 3)(2x + 5)\)[/tex].

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