To determine which terms, when added to [tex]\(3x^2y\)[/tex], will result in a monomial, let's analyze each given term one by one:
1. Term: [tex]\(3xy\)[/tex]
- [tex]\(3x^2y\)[/tex] has the variables [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex].
- [tex]\(3xy\)[/tex] has the variables [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
- The terms do not share the same degree of [tex]\(x\)[/tex], hence they cannot be combined to form a monomial.
- [tex]\((\boxed{Not~a~monomial})\)[/tex]
2. Term: [tex]\(-12x^2y\)[/tex]
- [tex]\(3x^2y\)[/tex] has exactly the same variables ([tex]\(x^2\)[/tex] and [tex]\(y\)[/tex]) as [tex]\(-12x^2y\)[/tex].
- Adding these terms will result in [tex]\((3 - 12)x^2y = -9x^2y\)[/tex], which is a monomial.
- [tex]\((\boxed{Valid~monomial})\)[/tex]
3. Term: [tex]\(2x^2y^2\)[/tex]
- [tex]\(3x^2y\)[/tex] includes the variables [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex].
- [tex]\(2x^2y^2\)[/tex] includes the variables [tex]\(x^2\)[/tex] and [tex]\(y^2\)[/tex].
- The terms have different degrees of [tex]\(y\)[/tex], hence they cannot be combined to form a monomial.
- [tex]\((\boxed{Not~a~monomial})\)[/tex]
4. Term: [tex]\(7xy^2\)[/tex]
- [tex]\(3x^2y\)[/tex] has the variables [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex].
- [tex]\(7xy^2\)[/tex] has the variables [tex]\(x\)[/tex] and [tex]\(y^2\)[/tex].
- The terms do not share the same degree of variables, hence they cannot be combined to form a monomial.
- [tex]\((\boxed{Not~a~monomial})\)[/tex]
5. Term: [tex]\(-10x^2\)[/tex]
- [tex]\(3x^2y\)[/tex] includes [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex].
- [tex]\(-10x^2\)[/tex] includes only [tex]\(x^2\)[/tex].
- The terms do not share the same variables, hence they cannot be combined to form a monomial.
- [tex]\((\boxed{Not~a~monomial})\)[/tex]
6. Term: [tex]\(4x^2y\)[/tex]
- [tex]\(3x^2y\)[/tex] includes the same variables [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex] as [tex]\(4x^2y\)[/tex].
- Adding these terms will result in [tex]\((3 + 4)x^2y = 7x^2y\)[/tex], which is a monomial.
- [tex]\((\boxed{Valid~monomial})\)[/tex]
7. Term: [tex]\(3x^3\)[/tex]
- [tex]\(3x^2y\)[/tex] has the variables [tex]\(x^2\)[/tex] and [tex]\(y\)[/tex].
- [tex]\(3x^3\)[/tex] only has the variable [tex]\(x^3\)[/tex].
- The terms do not share the same degree of variables, hence they cannot be combined to form a monomial.
- [tex]\((\boxed{Not~a~monomial})\)[/tex]
In conclusion, the terms that will result in a monomial when added to [tex]\(3x^2y\)[/tex] are:
- [tex]\(-12x^2y\)[/tex]
- [tex]\(4x^2y\)[/tex]
Thus, the valid terms are:
- [tex]\(-12 x^2 y\)[/tex]
- [tex]\(4 x^2 y\)[/tex]
