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We are tasked with factoring the quadratic expression \( z^2 + 10z + 21 \).

### Steps to Factor:

We need to find two numbers that multiply to \( 21 \) (the constant term) and add up to \( 10 \) (the coefficient of \( z \)).

- The factors of \( 21 \) are: \( (1, 21), (3, 7) \).
- Among these, \( 3 + 7 = 10 \).

Thus, the expression can be factored as:

\[
z^2 + 10z + 21 = (z + 3)(z + 7)
\]

### Final Answer:
The factored expression is \( (z + 3)(z + 7) \).

Would you like further details or explanations on any part of this?

Here are 5 related questions to expand on this:

1. How do you factor a quadratic expression with different coefficients?
2. What is the relationship between factoring and solving quadratic equations?
3. How can we verify that our factorization is correct?
4. Can this method be used for quadratic equations with negative terms?
5. What if the quadratic expression cannot be factored easily by simple inspection?

**Tip**: When factoring quadratics, always check the product and sum of the constant term and the linear term to ensure correctness.

Learn how to factor the quadratic expression z^2 + 10z + 21 step-by-step. This algebra problem demonstrates how to identify factors of the constant term and find numbers that add up to the coefficient of the linear term. Suitable for Grades 7-9.

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