Let's determine which of the given expressions is equal to the polynomial [tex]\(2x^4 + 5x^3 - 8x - 20\)[/tex].
### Option 1: [tex]\(\mathbf{x^3(2x + 5) + 4(2x - 5)}\)[/tex]
First, we expand the terms individually:
[tex]\[ x^3(2x + 5) = x^3 \cdot 2x + x^3 \cdot 5 = 2x^4 + 5x^3 \][/tex]
[tex]\[ 4(2x - 5) = 4 \cdot 2x + 4 \cdot (-5) = 8x - 20 \][/tex]
Combining these results:
[tex]\[ 2x^4 + 5x^3 + 8x - 20 \][/tex]
### Option 2: [tex]\(\mathbf{x^3(2x + 5) - 4(2x + 5)}\)[/tex]
First, we expand the terms individually:
[tex]\[ x^3(2x + 5) = x^3 \cdot 2x + x^3 \cdot 5 = 2x^4 + 5x^3 \][/tex]
[tex]\[ -4(2x + 5) = -4 \cdot 2x - 4 \cdot 5 = -8x - 20 \][/tex]
Combining these results:
[tex]\[ 2x^4 + 5x^3 - 8x - 20 \][/tex]
### Option 3: [tex]\(\mathbf{x^3(2x - 5) - 4(2x - 5)}\)[/tex]
First, we expand the terms individually:
[tex]\[ x^3(2x - 5) = x^3 \cdot 2x - x^3 \cdot 5 = 2x^4 - 5x^3 \][/tex]
[tex]\[ -4(2x - 5) = -4 \cdot 2x + 4 \cdot 5 = -8x + 20 \][/tex]
Combining these results:
[tex]\[ 2x^4 - 5x^3 - 8x + 20 \][/tex]
### Option 4: [tex]\(\mathbf{x^3(2x + 5) + 4(2x + 5)}\)[/tex]
First, we expand the terms individually:
[tex]\[ x^3(2x + 5) = x^3 \cdot 2x + x^3 \cdot 5 = 2x^4 + 5x^3 \][/tex]
[tex]\[ 4(2x + 5) = 4 \cdot 2x + 4 \cdot 5 = 8x + 20 \][/tex]
Combining these results:
[tex]\[ 2x^4 + 5x^3 + 8x + 20 \][/tex]
### Conclusion
Comparing each expanded polynomial to the given polynomial [tex]\(2x^4 + 5x^3 - 8x - 20\)[/tex]:
- Option 1 results in: [tex]\(2x^4 + 5x^3 + 8x - 20\)[/tex]
- Option 2 results in: [tex]\(2x^4 + 5x^3 - 8x - 20\)[/tex] (The same as given polynomial)
- Option 3 results in: [tex]\(2x^4 - 5x^3 - 8x + 20\)[/tex]
- Option 4 results in: [tex]\(2x^4 + 5x^3 + 8x + 20\)[/tex]
Therefore, the expression that is equal to the polynomial [tex]\(2x^4 + 5x^3 - 8x - 20\)[/tex] is:
[tex]\[ \boxed{x^3(2 x+5) - 4(2 x+5)} \][/tex] or simply Option 2.
