To solve the quadratic equation $4x^2 - 10x = 0$ by factoring, follow these steps:

1. Set the equation to zero:
   The equation is already set to zero:
   $4x^2 - 10x = 0$

2. Factor out the greatest common factor (GCF):
   Identify the GCF of the terms $4x^2$ and $-10x$. The GCF is $2x$.
   Factor $2x$ out of the equation:
   $2x(2x - 5) = 0$

3. Apply the zero product property:
   According to the zero product property, if a product of two factors is zero, at least one of the factors must be zero. Set each factor equal to zero:
   $2x = 0 \quad \text{or} \quad 2x - 5 = 0$

4. Solve each equation:
   Solve the first equation:
   $2x = 0$
   $x = 0$

   Solve the second equation:
   $2x - 5 = 0$
   $2x = 5$
   $x = \frac{5}{2}$

So, the solutions to the quadratic equation $4x^2 - 10x = 0$ are: $x = 0 \quad \text{and} \quad x = \frac{5}{2}$

Would you like more details or have any questions about this solution?

Here are 8 related questions for further exploration:

1. How do you factor quadratic equations in general?
2. What are other methods to solve quadratic equations?
3. Can you explain the zero product property in more detail?
4. How do you solve quadratic equations using the quadratic formula?
5. How do you complete the square to solve quadratic equations?
6. What are some applications of quadratic equations in real life?
7. Can all quadratic equations be factored easily?
8. How do you determine the nature of the roots of a quadratic equation?

Tip: Always check your factored form by expanding it back out to ensure it matches the original equation. This helps verify that you factored it correctly.