To determine the possible dimensions of the given cuboid with a volume of [tex]\( 5x^2 - 125 \)[/tex], we need to check which set of dimensions, when multiplied together, will give us this polynomial expression for the volume.
Given volume: [tex]\[ 5x^2 - 125 \][/tex]
### Checking Option 1: [tex]\((x + 5), (x - 5), 1\)[/tex]
1. Calculate the product of the dimensions:
[tex]\[ (x + 5) \times (x - 5) \times 1 \][/tex]
2. Simplify the product:
[tex]\[ (x + 5)(x - 5) \times 1 = (x^2 - 25) \times 1 = x^2 - 25 \][/tex]
3. Compare with the given volume:
[tex]\[ x^2 - 25 \neq 5x^2 - 125 \][/tex]
Thus, Option 1 is not correct.
### Checking Option 2: [tex]\(3, 3, 3\)[/tex]
1. Calculate the product of the dimensions:
[tex]\[ 3 \times 3 \times 3 \][/tex]
2. Simplify the product:
[tex]\[ 3 \times 3 \times 3 = 27 \][/tex]
3. Compare with the given volume:
[tex]\[ 27 \neq 5x^2 - 125 \][/tex]
Thus, Option 2 is not correct.
### Checking Option 3: [tex]\(5, (x^2 - 125)\)[/tex]
1. Calculate the product of the dimensions:
[tex]\[ 5 \times (x^2 - 125) \][/tex]
2. Simplify the product:
[tex]\[ 5 \times (x^2 - 125) = 5x^2 - 625 \][/tex]
3. Compare with the given volume:
[tex]\[ 5x^2 - 625 \neq 5x^2 - 125 \][/tex]
Thus, Option 3 is not correct.
### Checking Option 4: [tex]\(5, (x + 5), (x - 5)\)[/tex]
1. Calculate the product of the dimensions:
[tex]\[ 5 \times (x + 5) \times (x - 5) \][/tex]
2. Simplify the product:
[tex]\[ 5 \times (x^2 - 25) = 5x^2 - 125 \][/tex]
3. Compare with the given volume:
[tex]\[ 5x^2 - 125 = 5x^2 - 125 \][/tex]
Thus, Option 4 is correct.
### Conclusion
The correct set of dimensions for the cuboid, given its volume as [tex]\( 5x^2 - 125 \)[/tex], is:
[tex]\[ \boxed{5, (x + 5), (x - 5)} \][/tex]
