https://file.aiquickdraw.com/mathbot/file/17249233876924swff6qx.jpg

The equation in the image is:

\[
\frac{x^2 + 1}{x^2 - 3} + \frac{2x^2}{x^2 - 3} = 4
\]

To find the value of \(x\), let's solve the equation step-by-step.

### Step-by-step Solution:

1. **Combine the fractions**:
   Since both fractions have the same denominator, \(x^2 - 3\), we can combine them:

   \[
   \frac{x^2 + 1 + 2x^2}{x^2 - 3} = 4
   \]

   Simplify the numerator:
   \[
   \frac{3x^2 + 1}{x^2 - 3} = 4
   \]

2. **Cross-multiply**:
   \[
   3x^2 + 1 = 4(x^2 - 3)
   \]

   Expand the right side:
   \[
   3x^2 + 1 = 4x^2 - 12
   \]

3. **Rearrange the equation**:
   Move all terms to one side:
   \[
   3x^2 + 1 - 4x^2 + 12 = 0
   \]

   Simplify:
   \[
   -x^2 + 13 = 0
   \]

   Which is equivalent to:
   \[
   x^2 = 13
   \]

4. **Solve for \(x\)**:
   Taking the square root of both sides, we get:

   \[
   x = \sqrt{13} \quad \text{or} \quad x = -\sqrt{13}
   \]

### Solutions:
\[
x = \sqrt{13} \text{ or } x = -\sqrt{13}
\]

Both values of \(x\) are valid as they do not make the denominator zero.

Would you like more details or have any questions?

### Related Questions:
1. What are the steps to simplify a rational expression?
2. Why is it necessary to check for possible restrictions in the denominator?
3. How do you handle equations with higher-degree polynomials in the numerator and denominator?
4. Can you explain the process of cross-multiplication in rational equations?
5. What are some strategies for solving quadratic equations involving square roots?

### Tip:
Always check that the values you find for \(x\) do not make any denominators zero, which would make the original equation undefined!

Explore how to find the value of x in a complex rational equation step-by-step. This detailed solution covers combining fractions, cross-multiplication, and solving quadratic equations. Suitable for advanced high school students.

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