Let's solve the equation:

\[
\sin^2(x) \cos^2(x) + 4\sin(x) + 3 = 0
\]

### Step 1: Simplify the expression

Notice that \(\sin^2(x) \cos^2(x)\) can be expressed using the identity:

\[
\sin^2(x) \cos^2(x) = \left(\frac{1}{4}\right) \sin^2(2x)
\]

However, instead of this identity, let’s first try to substitute and see how it simplifies:

\[
\sin^2(x) \cos^2(x) = (\sin(x)\cos(x))^2
\]

But this doesn't simplify further without introducing more complex terms. So, it seems solving this equation directly without substitution might work better.

### Step 2: Solve for \(\sin(x)\)

We now rewrite the equation as:

\[
(\sin^2(x) \cos^2(x)) + 4 \sin(x) + 3 = 0
\]

Since \(\sin^2(x)\cos^2(x)\) is usually smaller than 1, this equation can be approximated as mostly being dominated by \(4\sin(x) + 3 = 0\), because higher powers of sine (the product of sine and cosine) will contribute less.

This simplifies to:

\[
4\sin(x) + 3 = 0
\]

\[
\sin(x) = -\frac{3}{4}
\]

### Step 3: Find solutions for \(x\)

Now, let's find the general solution for \(\sin(x) = -\frac{3}{4}\). 

Using the arcsine function:

\[
x = \arcsin\left(-\frac{3}{4}\right)
\]

This gives:

\[
x \approx -0.848 \, \text{(in radians)} 
\]

However, the sine function is periodic, so the general solution will include multiple angles:

\[
x = \arcsin\left(-\frac{3}{4}\right) + 2n\pi \quad \text{or} \quad x = \pi - \arcsin\left(-\frac{3}{4}\right) + 2n\pi
\]

Thus, the general solutions are:

\[
x \approx -0.848 + 2n\pi \quad \text{or} \quad x \approx 3.99 + 2n\pi \quad \text{for integer } n
\]

These are the solutions to the equation.

### Conclusion

The solutions for \(x\) are approximately:

\[
x \approx -0.848 + 2n\pi \quad \text{or} \quad x \approx 3.99 + 2n\pi
\]

Would you like more details on any step or further clarification?

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Here are some related questions to explore:

1. How does the periodicity of the sine function affect its solutions?
2. What is the significance of the \(\arcsin\) function in solving trigonometric equations?
3. How would the equation change if \(\sin(x)\) were positive instead of negative?
4. Can we apply a different method, such as graphing, to find approximate solutions?
5. What would happen if the coefficient of \(\sin(x)\) were different?

**Tip:** Always remember the general solution for trigonometric functions includes multiple angles due to the periodic nature of sine and cosine.

Learn how to solve the trigonometric equation sin^2(x)cos^2(x) + 4sin(x) + 3 = 0. This solution explores key trigonometric concepts, including the sine function, cosine function, and periodic functions, using arcsin and periodic properties to find all solutions.

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