To solve the equation \((x + 1)(3x - 5) = 0\), we can use the zero product property, which states that if the product of two factors is zero, at least one of the factors must be zero.

This gives us two separate equations to solve:

1. \(x + 1 = 0\)
2. \(3x - 5 = 0\)

**Solving the first equation:**
\[
x + 1 = 0 \implies x = -1
\]

**Solving the second equation:**
\[
3x - 5 = 0 \implies 3x = 5 \implies x = \frac{5}{3}
\]

Thus, the solutions to the equation \((x + 1)(3x - 5) = 0\) are:
\[
x = -1 \quad \text{and} \quad x = \frac{5}{3}
\]