solve the quadratic equation 84x^2+16x=0 by factoring out the GCF

To solve the quadratic equation \( 84x^2 + 16x = 0 \) by factoring out the greatest common factor (GCF), we first identify the GCF of the terms:

1. The GCF of \( 84x^2 \) and \( 16x \) is \( 4x \).
2. Factoring out \( 4x \) from the equation gives us:

\[
4x(21x + 4) = 0
\]

Next, we set each factor equal to zero:

1. From \( 4x = 0 \):
\[
x = 0
\]

2. From \( 21x + 4 = 0 \):
\[
21x = -4 \implies x = -\frac{4}{21}
\]

Therefore, the solution set is:

\[
x = \left\{ 0, -\frac{4}{21} \right\}
\]

So the correct answer from the options you provided is:

**The solution set is** \( x = \left\{ -\frac{4}{21}, 0 \right\} \).