To factor the polynomial [tex]\(2x^4 + 5x^3 - 8x - 20\)[/tex], let's go through the process step by step until we reach the correct factorization.
### Step 1: Look for Common Factors
First, we need to see if there is an obvious common factor for the entire expression. Here, there are no common factors among all the terms, so we proceed to the next steps.
### Step 2: Grouping Terms
Then, we can consider grouping terms or using other factoring techniques like synthetic division or finding roots. However, this polynomial does not lend itself easily to grouping straightforwardly, so we need to think of a factorization method that goes deeper.
### Step 3: Factorization
Given that we have no obvious initial grouping, let's consider that the polynomial can be factored into a product of simpler polynomials of lower degree.
We have the polynomial:
[tex]\[ 2x^4 + 5x^3 - 8x - 20 \][/tex]
To hypothesize potential factorizations, we test a few simpler lower-degree polynomial forms. With trial and error or using more advanced algebraic techniques, we discover:
The polynomial can be factored as:
[tex]\[ (2x + 5)(x^3 - 4) \][/tex]
### Verification
Let's verify that this indeed works by expanding:
[tex]\[
(2x + 5)(x^3 - 4)
\][/tex]
Distribute [tex]\(2x + 5\)[/tex] to each term in [tex]\(x^3 - 4\)[/tex]:
[tex]\[
= (2x + 5) \cdot x^3 + (2x + 5) \cdot (-4)
\][/tex]
[tex]\[
= 2x \cdot x^3 + 5 \cdot x^3 + 2x \cdot (-4) + 5 \cdot (-4)
\][/tex]
[tex]\[
= 2x^4 + 5x^3 - 8x - 20
\][/tex]
### Final Answer
Thus, upon verifying, we indeed have the correct factorization:
[tex]\[
(2x + 5)(x^3 - 4)
\][/tex]
This confirms that the polynomial [tex]\(2x^4 + 5x^3 - 8x - 20\)[/tex] can be factored into:
[tex]\[
\boxed{(2x + 5)(x^3 - 4)}
\][/tex]
