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We are given the equation to solve:

\[
\sqrt{3x - 2} - \sqrt{x + 1} = 3
\]

### Steps to solve:

1. **Isolate one square root**:
   Move one of the square root terms to the other side:
   \[
   \sqrt{3x - 2} = 3 + \sqrt{x + 1}
   \]

2. **Square both sides**:
   Squaring both sides to eliminate the square root on the left-hand side:
   \[
   (\sqrt{3x - 2})^2 = (3 + \sqrt{x + 1})^2
   \]
   This simplifies to:
   \[
   3x - 2 = (3)^2 + 2(3)(\sqrt{x+1}) + (\sqrt{x+1})^2
   \]
   \[
   3x - 2 = 9 + 6\sqrt{x+1} + (x + 1)
   \]
   \[
   3x - 2 = 10 + x + 6\sqrt{x + 1}
   \]

3. **Isolate the remaining square root**:
   Simplify the equation and isolate the square root term:
   \[
   3x - x - 2 - 10 = 6\sqrt{x + 1}
   \]
   \[
   2x - 12 = 6\sqrt{x + 1}
   \]
   Divide by 6:
   \[
   \frac{2x - 12}{6} = \sqrt{x + 1}
   \]
   Simplify:
   \[
   \frac{x - 6}{3} = \sqrt{x + 1}
   \]

4. **Square both sides again**:
   Square both sides again to eliminate the remaining square root:
   \[
   \left(\frac{x - 6}{3}\right)^2 = (\sqrt{x + 1})^2
   \]
   \[
   \frac{(x - 6)^2}{9} = x + 1
   \]
   Multiply both sides by 9:
   \[
   (x - 6)^2 = 9(x + 1)
   \]
   Expand both sides:
   \[
   x^2 - 12x + 36 = 9x + 9
   \]

5. **Solve the quadratic equation**:
   Move all terms to one side:
   \[
   x^2 - 12x + 36 - 9x - 9 = 0
   \]
   \[
   x^2 - 21x + 27 = 0
   \]
   Solve this quadratic using the quadratic formula:
   \[
   x = \frac{-(-21) \pm \sqrt{(-21)^2 - 4(1)(27)}}{2(1)}
   \]
   \[
   x = \frac{21 \pm \sqrt{441 - 108}}{2}
   \]
   \[
   x = \frac{21 \pm \sqrt{333}}{2}
   \]

Thus, the exact solutions are:

\[
x = \frac{21 \pm \sqrt{333}}{2}
\]

Let me know if you'd like me to simplify this further or calculate the decimal approximations!

Would you like to discuss any steps in more detail? Here are 5 questions for further exploration:

1. How do you solve other equations with multiple square roots?
2. What does it mean when you square both sides of an equation?
3. How do you apply the quadratic formula to different types of quadratics?
4. What is the discriminant and how does it tell you the nature of the roots?
5. Why is it important to check solutions in equations involving square roots?

**Tip:** After squaring both sides of an equation, always check for extraneous solutions, as squaring can introduce invalid answers.

This page provides a step-by-step solution to the radical equation √(3x - 2) - √(x + 1) = 3. Learn how to isolate the square roots, square both sides, and solve the resulting quadratic equation. Suitable for high school students, this solution highlights key algebraic concepts and the quadratic formula.

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