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We are given the function \( h(x) = 12x^2 - (n+22)x + p \), and we know:

1. The graph of \( h(x) \) intersects the y-axis at \( (0, -5) \).
2. A factor of \( h(x) \) is \( x - 4 \).

### Step-by-step solution:

1. **Using the intersection point \( (0, -5) \):**

   The y-intercept is the point where \( x = 0 \). At this point, \( h(0) = -5 \). Substituting into the equation:
   \[
   h(0) = 12(0)^2 - (n+22)(0) + p = p
   \]
   Therefore, \( p = -5 \).

2. **Using the factor \( x - 4 \):**

   Since \( x - 4 \) is a factor of \( h(x) \), we know that \( h(4) = 0 \). Substituting \( x = 4 \) into the function:
   \[
   h(4) = 12(4)^2 - (n+22)(4) + p = 0
   \]
   Now, substitute \( p = -5 \) into the equation:
   \[
   h(4) = 12(16) - (n+22)(4) - 5 = 0
   \]
   Simplifying:
   \[
   192 - 4(n + 22) - 5 = 0
   \]
   \[
   192 - 4n - 88 - 5 = 0
   \]
   \[
   99 - 4n = 0
   \]
   Solving for \( n \):
   \[
   4n = 99 \quad \Rightarrow \quad n = \frac{99}{4} = 24.75
   \]

### Final answer:
- \( n = 24.75 \)
- \( p = -5 \)

Would you like further details or have any questions?

Here are some related questions:
1. What happens if a quadratic equation has complex roots?
2. How does the graph of a quadratic equation change with different values of \( n \) and \( p \)?
3. How do we factor a quadratic expression using the roots?
4. Can a quadratic function have more than two real roots?
5. What does the discriminant of a quadratic equation tell us about its roots?

**Tip:** To factor quadratic equations, always check for common factors or known roots, and use techniques like completing the square or the quadratic formula.

Given the function h(x) = 12x^2 - (n + 22)x + p, and knowing that the graph of h intersects the y-axis at (0, -5), and one factor of h(x) is x - 4, determine the values of the constants n and p.

Learn how to solve for constants in a quadratic function using given points and factor information. The problem involves key concepts like quadratic functions, factoring, and the Factor Theorem, and is suitable for students in Grades 10-12.

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