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The equation presented is:

\[
(3z + 4)(8 + z) = 0
\]

To solve this equation, we can apply the **Zero Product Property**, which states that if the product of two factors is zero, then at least one of the factors must be zero. Therefore, we set each factor equal to zero and solve for \( z \) individually.

1. Set \( 3z + 4 = 0 \):
   \[
   3z + 4 = 0
   \]
   Subtract 4 from both sides:
   \[
   3z = -4
   \]
   Divide by 3:
   \[
   z = -\frac{4}{3}
   \]

2. Set \( 8 + z = 0 \):
   \[
   8 + z = 0
   \]
   Subtract 8 from both sides:
   \[
   z = -8
   \]

Thus, the solutions are:
\[
z = -\frac{4}{3}, -8
\]

Would you like a deeper explanation on any step or have questions?

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Here are some follow-up questions:
1. What is the Zero Product Property, and why does it apply here?
2. How would the solution change if the equation were \((3z + 4)(8 + z) = 5\) instead of zero?
3. Can we solve this type of problem using graphing methods? How?
4. What other properties of equations can be used to solve similar problems?
5. Why is it necessary to check both factors separately in this type of equation?

**Tip:** Always check if a product equals zero when solving equations with factors, as this often simplifies the solution process.

Solve (3z + 4)(8 + z) = 0. (If there is more than one solution, separate them with commas.)

Learn how to solve the equation (3z + 4)(8 + z) = 0 using the Zero Product Property. This problem is ideal for students in Grades 8-10, covering algebraic concepts and factoring techniques.

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