Let's determine which of the given numbers are irrational one by one:
1. [tex]\(-4.8237\)[/tex]:
- This is a terminating decimal number.
- Terminating decimal numbers can be expressed as a fraction of two integers (i.e., a rational number).
- Therefore, [tex]\(-4.8237\)[/tex] is a rational number.
2. [tex]\(\frac{\pi}{2}\)[/tex]:
- [tex]\(\pi\)[/tex] (pi) itself is an irrational number, which cannot be written as a fraction of two integers.
- When an irrational number ([tex]\(\pi\)[/tex]) is divided by a non-zero rational number (in this case, 2), the result is still an irrational number.
- Therefore, [tex]\(\frac{\pi}{2}\)[/tex] is an irrational number.
3. [tex]\(\sqrt[3]{4}\)[/tex]:
- The cube root of 4 ([tex]\(\sqrt[3]{4}\)[/tex]) is not a perfect cube, meaning it cannot be expressed as a fraction of two integers.
- Therefore, [tex]\(\sqrt[3]{4}\)[/tex] is an irrational number.
4. [tex]\(4 + \sqrt{25}\)[/tex]:
- [tex]\(\sqrt{25}\)[/tex] is the square root of 25, which is 5, a whole number.
- Adding a whole number to another whole number yields a rational number (e.g., [tex]\(4 + 5 = 9\)[/tex]).
- Therefore, [tex]\(4 + \sqrt{25} = 9\)[/tex], which is a rational number.
From this analysis, the irrational numbers among the given choices are:
[tex]\[
\frac{\pi}{2} \text{ and } \sqrt[3]{4}
\][/tex]
Thus, the correct selection is:
[tex]\[
\boxed{2}
\][/tex]
