Let's carefully examine the given options to determine which term is not a like term with the others.
### List of Terms:
1. [tex]\(-3xy\)[/tex]
2. [tex]\(xy\)[/tex]
3. [tex]\(8yx\)[/tex]
4. [tex]\(5x^2y\)[/tex]
#### Definition of Like Terms:
Like terms are terms that contain the same variables raised to the same power. The coefficients of the terms can be different, but the variable parts must be identical.
### Analysis:
1. [tex]\(-3xy\)[/tex]:
- Variables: [tex]\(x\)[/tex] and [tex]\(y\)[/tex]
- Powers: [tex]\(x^1y^1\)[/tex]
2. [tex]\(xy\)[/tex]:
- Variables: [tex]\(x\)[/tex] and [tex]\(y\)[/tex]
- Powers: [tex]\(x^1y^1\)[/tex]
3. [tex]\(8yx\)[/tex]:
- Variables: [tex]\(x\)[/tex] and [tex]\(y\)[/tex]
- Powers: Here, the order of multiplication makes no difference (since [tex]\(8yx\)[/tex] is the same as [tex]\(8xy\)[/tex]):
- Powers: [tex]\(x^1y^1\)[/tex]
4. [tex]\(5x^2y\)[/tex]:
- Variables: [tex]\(x\)[/tex] and [tex]\(y\)[/tex]
- Powers: [tex]\(x^2y^1\)[/tex]
### Conclusion:
- The first three terms ([tex]\(-3xy\)[/tex], [tex]\(xy\)[/tex], [tex]\(8yx\)[/tex]) all contain the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex] raised to the same powers ([tex]\(x^1y^1\)[/tex]).
- The fourth term ([tex]\(5x^2y\)[/tex]) contains [tex]\(x\)[/tex] raised to the power of 2 and [tex]\(y\)[/tex] raised to the power of 1 ([tex]\(x^2y^1\)[/tex]).
Since [tex]\(5x^2y\)[/tex] has a different power for [tex]\(x\)[/tex] compared to the others, it is not a like term with the rest.
Thus, the term that is NOT a like term with the others is:
- Option (d): [tex]\(5x^2y\)[/tex]
Therefore, the answer is [tex]\(4\)[/tex].
