To answer this question, let’s analyze each option based on the criteria provided: the expression must be a trinomial, have a leading coefficient of 3, and have a constant term of -5.
1. Option (A): [tex]\( 3m^2 - 5m + 1 \)[/tex]
- Number of terms: This expression has three terms, so it is a trinomial.
- Leading coefficient: The leading coefficient is the coefficient of the highest degree term [tex]\(3m^2\)[/tex], which is 3.
- Constant term: The constant term is the term without a variable, which is 1.
Thus, although this is a trinomial with the correct leading coefficient, the constant term does not match (-5).
2. Option (B): [tex]\( -5m^2 + 4m + 3 \)[/tex]
- Number of terms: This expression has three terms, so it is a trinomial.
- Leading coefficient: The leading coefficient is the coefficient of the highest degree term [tex]\(-5m^2\)[/tex], which is -5.
- Constant term: The constant term is 3.
Although this is a trinomial, neither the leading coefficient nor the constant term match the given criteria.
3. Option (C): [tex]\( 3m^2 - 5 \)[/tex]
- Number of terms: This expression has two terms, so it is not a trinomial.
- Leading coefficient: The leading coefficient is 3.
- Constant term: The constant term is -5.
While the leading coefficient and constant term match the given criteria, this is not a trinomial.
4. Option (D): [tex]\( 3m^2 + m - 5 \)[/tex]
- Number of terms: This expression has three terms, so it is a trinomial.
- Leading coefficient: The leading coefficient is the coefficient of the highest degree term [tex]\(3m^2\)[/tex], which is 3.
- Constant term: The constant term is -5.
This expression meets all the given criteria: it is a trinomial with a leading coefficient of 3 and a constant term of -5.
Therefore, the correct answer is:
(D) [tex]\( 3m^2 + m - 5 \)[/tex]
