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To factor the trinomial \( m^2 + 12m + 35 \), we need to find two numbers that multiply to \( 35 \) and add up to \( 12 \).

1. The factors of \( 35 \) are:
   - \( 1 \times 35 \)
   - \( 5 \times 7 \)

2. The pair \( 5 \) and \( 7 \) add up to \( 12 \), so we can use them to factor the expression.

The factored form is:
\[
m^2 + 12m + 35 = (m + 5)(m + 7)
\]

Therefore, the factors are \( (m + 5) \) and \( (m + 7) \).

Would you like further details on this factoring process, or have any questions?

Here are some related questions to deepen understanding:
1. How can we verify if the factorization is correct?
2. What if the middle term was negative; how would that change the factors?
3. What are the steps to factor a trinomial when the leading coefficient is not 1?
4. How can you identify a trinomial that is not factorable over the integers?
5. What other methods exist for factoring trinomials, like completing the square?

**Tip:** Always check the product of the factors to confirm that they multiply back to the original trinomial.

Learn how to factor the trinomial m^2 + 12m + 35 by finding two numbers that add to 12 and multiply to 35. Suitable for students in Grades 8-10, covering key algebraic concepts and factoring techniques.

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