Let's solve the equation \( 10x^2 + 2x = 0 \).

### Step 1: Factor the equation
We can factor out the common factor, which is \( 2x \), from both terms:
\[
2x(5x + 1) = 0
\]

### Step 2: Solve each factor
Now, set each factor equal to zero and solve for \( x \):

1. \( 2x = 0 \)
   \[
   x = 0
   \]

2. \( 5x + 1 = 0 \)
   \[
   5x = -1 \implies x = -\frac{1}{5}
   \]

### Final Answer:
The solutions to the equation are:
\[
x = 0 \quad \text{or} \quad x = -\frac{1}{5}
\]

Would you like more details or have any questions about this solution? Here are some related questions for practice:

1. Solve \( 5x^2 + 3x = 0 \).
2. Factor and solve \( 12x^2 + 8x = 0 \).
3. What is the solution to \( 7x^2 + 14x = 0 \)?
4. How do you solve \( x(4x + 9) = 0 \)?
5. Find the roots of \( 3x^2 + 9x = 0 \).

**Tip:** Always check if you can factor an equation to simplify the solution process.

This step-by-step guide shows how to solve the quadratic equation 10x^2 + 2x = 0. Learn factoring and how to apply the Zero Product Property to find the solutions. This problem is ideal for students in Grades 8-10, covering important algebraic concepts.

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