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The given quadratic expression is \( h^2 + 6h - 27 \). To factor this, we look for two numbers that multiply to give the constant term \(-27\) and add to give the middle term \(6\).

Let's proceed step-by-step:

1. The product of the first and last term is \( -27 \).
2. We need two numbers that multiply to \( -27 \) and add to \( 6 \). These numbers are \( 9 \) and \( -3 \), because:
   \[
   9 \times (-3) = -27 \quad \text{and} \quad 9 + (-3) = 6.
   \]
3. We can now rewrite the middle term using these numbers:
   \[
   h^2 + 9h - 3h - 27.
   \]
4. Group the terms:
   \[
   (h^2 + 9h) - (3h + 27).
   \]
5. Factor each group:
   \[
   h(h + 9) - 3(h + 9).
   \]
6. Now, factor out the common binomial:
   \[
   (h - 3)(h + 9).
   \]

Thus, the factored expression is \( (h - 3)(h + 9) \).

Would you like more details or have any questions? 

Here are 5 related questions to expand your understanding:
1. What is the significance of factoring quadratic expressions?
2. How can the zero-product property be used after factoring a quadratic equation?
3. How do you recognize when a quadratic expression is factorable?
4. Can every quadratic expression be factored into real numbers?
5. What is the connection between the discriminant and factoring quadratics?

**Tip:** When factoring, always check if you can simplify using common factors before applying other methods like splitting the middle term.

Factor h^2 + 6h - 27. The factored expression is ____.

Learn how to factor the quadratic expression h^2 + 6h - 27 using middle term splitting and zero product property. This problem covers essential algebraic concepts and is suitable for Grades 8-10.

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