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To solve the quadratic equation \(84x^2 + 16x = 0\) by factoring out the greatest common factor (GCF), let's first identify the GCF of the coefficients 84 and 16.
The GCF of 84 and 16 is 4. We can factor out 4 from the equation:
\[
84x^2 + 16x = 4(21x^2 + 4x) = 0
\]
Now, we can set the equation inside the parentheses to zero:
\[
21x^2 + 4x = 0
\]
Next, we can factor out an \(x\):
\[
x(21x + 4) = 0
\]
Setting each factor equal to zero gives us the solutions:
1. \(x = 0\)
2. \(21x + 4 = 0\) which leads to \(21x = -4\) or \(x = -\frac{4}{21}\)
Thus, the solutions to the equation are:
\[
x = 0 \quad \text{and} \quad x = -\frac{4}{21}
\]
So, the correct response is:
**The solutions are \(x = 0\) and \(x = -\frac{4}{21}\).**