based on the zero product property which of the following statements must be true about the quadratic equation (3x+4)(x-2)=0

The zero product property states that if the product of two expressions equals zero, then at least one of the expressions must be equal to zero.

For the quadratic equation given, \((3x + 4)(x - 2) = 0\), we can apply the zero product property. This means:

1. \(3x + 4 = 0\) or
2. \(x - 2 = 0\)

Now, let’s solve each equation:

1. For \(3x + 4 = 0\):
\[
3x = -4 \\
x = -\frac{4}{3}
\]

2. For \(x - 2 = 0\):
\[
x = 2
\]

So, the solutions to the equation \((3x + 4)(x - 2) = 0\) are \(x = -\frac{4}{3}\) and \(x = 2\).

Therefore, the statements that must be true about the quadratic equation are:
- \(x = -\frac{4}{3}\) is a solution.
- \(x = 2\) is a solution.

In conclusion, the solutions of the equation \( (3x + 4)(x - 2) = 0 \) are \(x = -\frac{4}{3}\) and \(x = 2\).