To solve the equation \( -2\sin\left(\frac{3x}{2}\right)\sin\left(\frac{x}{2}\right) - \sin(3x) = 0 \), let's go through the following steps:

### Step 1: Factor the equation
The equation can be factored by recognizing that \( \sin(3x) \) can be expressed as a product involving \( \sin\left(\frac{3x}{2}\right) \).

The equation is:

\[
-2\sin\left(\frac{3x}{2}\right)\sin\left(\frac{x}{2}\right) - \sin(3x) = 0
\]

Notice that \( \sin(3x) \) can be expressed using the triple-angle identity:

\[
\sin(3x) = 3\sin(x) - 4\sin^3(x)
\]

However, in this case, it’s more convenient to factor directly:

\[
\sin(3x) = 2\sin\left(\frac{3x}{2}\right)\cos\left(\frac{3x}{2}\right)
\]

Substituting this into the equation:

\[
-2\sin\left(\frac{3x}{2}\right)\sin\left(\frac{x}{2}\right) - 2\sin\left(\frac{3x}{2}\right)\cos\left(\frac{3x}{2}\right) = 0
\]

### Step 2: Factor out common terms
Factor out \( -2\sin\left(\frac{3x}{2}\right) \):

\[
-2\sin\left(\frac{3x}{2}\right)\left[\sin\left(\frac{x}{2}\right) + \cos\left(\frac{3x}{2}\right)\right] = 0
\]

This equation will be zero if either factor is zero.

### Step 3: Solve each factor separately
We have two factors to consider:

1. \( \sin\left(\frac{3x}{2}\right) = 0 \)
2. \( \sin\left(\frac{x}{2}\right) + \cos\left(\frac{3x}{2}\right) = 0 \)

#### For \( \sin\left(\frac{3x}{2}\right) = 0 \):
The sine function is zero at multiples of \( \pi \):

\[
\frac{3x}{2} = n\pi, \quad \text{where } n \text{ is an integer}
\]

Solving for \( x \):

\[
x = \frac{2n\pi}{3}
\]

#### For \( \sin\left(\frac{x}{2}\right) + \cos\left(\frac{3x}{2}\right) = 0 \):
This is a more complex equation and requires solving:

\[
\sin\left(\frac{x}{2}\right) = -\cos\left(\frac{3x}{2}\right)
\]

This equation doesn't have a simple algebraic solution, so solutions would typically be found using numerical methods or by specific angle testing.

### Step 4: Consider the general solutions
For the first equation, we already have:

\[
x = \frac{2n\pi}{3}, \quad n \in \mathbb{Z}
\]

For the second equation, solutions depend on specific values of \( x \), so it might be necessary to test specific cases or use numerical approximation.

### Final Answer:
The general solution for the equation is:

\[
x = \frac{2n\pi}{3}, \quad n \in \mathbb{Z}
\]

And any other solutions from \( \sin\left(\frac{x}{2}\right) + \cos\left(\frac{3x}{2}\right) = 0 \) would need to be found numerically.

---

Do you have any questions or need further clarification? Here are some related topics you might find helpful:

1. What is the triple-angle identity, and how is it derived?
2. How do you solve trigonometric equations involving multiple factors?
3. What are the general solutions for trigonometric equations?
4. How can numerical methods be used to solve trigonometric equations?
5. What are the key identities for simplifying trigonometric expressions?

**Tip:** Factoring is a powerful method in solving trigonometric equations. Always check if the equation can be factored to simplify the solution process.

Learn how to solve the trigonometric equation -2sin(3x/2)sin(x/2)-sin(3x)=0 with detailed step-by-step solutions. This problem involves trigonometric identities and equations, including the triple-angle identity formula. Suitable for advanced high school students and those studying trigonometry.

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