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To solve the quadratic equation \(5x^2 + 75x = 0\) by factoring out the greatest common factor (GCF), we start by factoring out \(5x\):
\[
5x(x + 15) = 0
\]
Setting each factor equal to zero gives us the solutions:
1. \(5x = 0\) \(\Rightarrow x = 0\)
2. \(x + 15 = 0\) \(\Rightarrow x = -15\)
Thus, the solutions to the equation are:
\[
x = 0 \quad \text{and} \quad x = -15
\]
Therefore, the correct response is:
**The solutions are \(x = -15\) and \(x = 0\).** (This corresponds to the option: "The solutions are \(x = -15\) and \(x = 0\).")