To solve the quadratic equation \( 5x^2 + 75x = 0 \), we can start by factoring the equation.
First, factor out the common term \( 5x \):
\[ 5x(x + 15) = 0 \]
Now, set each factor equal to zero:
- \( 5x = 0 \)
- \( x + 15 = 0 \)
For the first equation:
\[ 5x = 0 \implies x = 0 \]
For the second equation:
\[ x + 15 = 0 \implies x = -15 \]
Thus, the solutions to the equation \( 5x^2 + 75x = 0 \) are
\[ \boxed{0} \quad \text{and} \quad \boxed{-15}. \]