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The given equation is:

\[
(a + 1)(a - 7) = 0
\]

This is a factored quadratic equation, and to solve for \(a\), we can apply the **zero product property**. The zero product property states that if the product of two factors is zero, then at least one of the factors must be zero.

So, set each factor equal to zero and solve for \(a\):

1. \( a + 1 = 0 \)
   
   \[
   a = -1
   \]

2. \( a - 7 = 0 \)
   
   \[
   a = 7
   \]

Thus, the solutions are \(a = -1\) and \(a = 7\).

If asked for multiple solutions, the answer should be written as:

\[
\boxed{-1, 7}
\]

Would you like more details or have any questions?

Here are 5 related questions you might explore:
1. How do you expand a factored quadratic expression?
2. What is the quadratic formula, and when is it used?
3. How would you solve the same equation graphically?
4. What happens if there is no real solution in a quadratic equation?
5. How do you complete the square to solve quadratic equations?

**Tip**: Always check your solutions by substituting them back into the original equation to ensure they satisfy it.

Learn how to solve quadratic equations by factoring with a detailed example. This guide covers the zero product property, a fundamental theorem in algebra used to find solutions to quadratic equations. Ideal for high school students learning algebra.

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