We start with the trinomial \( 3m^{2} + 10m + 21 \).
Step 2: Attempt to Factor the Polynomial
We check if the polynomial can be factored into the form \( (am + b)(cm + d) \).
Step 3: Analyze the Factorization
After attempting to factor, we find that the polynomial does not factor into simpler integer polynomials. The expression remains as \( 3m^{2} + 10m + 21 \).
Step 4: Conclusion on Factorization
Since the polynomial cannot be factored further, we conclude that it is prime. Thus, the polynomial \( 3m^{2} + 10m + 21 \) does not have a factorization over the integers.