To solve the quadratic equation $4x^2 + 7x = 0$, we can factorize it. Here's the step-by-step solution:

1. Start with the equation: $4x^2 + 7x = 0$

2. Factor out the common factor $x$: $x(4x + 7) = 0$

3. Set each factor equal to zero and solve for $x$: $x = 0 \quad \text{or} \quad 4x + 7 = 0$

4. Solve $4x + 7 = 0$:

   4x = -7 \\ x = -\frac{7}{4}$$

So, the solutions are: $x = 0 \quad \text{or} \quad x = -\frac{7}{4}$

Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. How do you solve quadratic equations by completing the square?
2. What is the quadratic formula and how is it derived?
3. How can you use the discriminant to determine the nature of the roots of a quadratic equation?
4. What are some real-life applications of quadratic equations?
5. How do you graph a quadratic equation and identify its key features?
6. What are the different methods for solving quadratic equations?
7. How can you solve quadratic equations using the method of substitution?
8. What is the relationship between the coefficients of a quadratic equation and its roots?

Tip: When solving quadratic equations, always check if you can factorize it easily before using more complex methods like the quadratic formula or completing the square.