Solving the Equation (5n  1)(n + 1) = 0 by Factoring
This equation is already factored for us, making it easy to solve. Here's how:
Understanding the Zero Product Property
The Zero Product Property states that if the product of two or more factors is zero, then at least one of the factors must be zero.
Applying the Zero Product Property

Set each factor equal to zero:
 5n  1 = 0
 n + 1 = 0

Solve for 'n' in each equation:
 5n = 1
 n = 1/5
 n = 1
Therefore, the solutions to the equation (5n  1)(n + 1) = 0 are n = 1/5 and n = 1.
Checking Our Solutions
We can check our solutions by substituting them back into the original equation:

For n = 1/5: (5(1/5)  1)(1/5 + 1) = (1  1)(6/5) = 0. This confirms n = 1/5 is a solution.

For n = 1: (5(1)  1)(1 + 1) = (6)(0) = 0. This confirms n = 1 is also a solution.
Conclusion
We have successfully solved the equation (5n  1)(n + 1) = 0 by factoring and utilizing the Zero Product Property. The solutions are n = 1/5 and n = 1.