To solve the equation 

\[
\sqrt{3v + 10} = \sqrt{3v + 3} + 1,
\]

we will follow these steps:

1. **Square both sides** to eliminate the square roots:
   \[
   (\sqrt{3v + 10})^2 = \left(\sqrt{3v + 3} + 1\right)^2.
   \]
   This gives:
   \[
   3v + 10 = (3v + 3) + 2\sqrt{3v + 3} + 1.
   \]
   Simplifying the right side results in:
   \[
   3v + 10 = 3v + 4 + 2\sqrt{3v + 3}.
   \]

2. **Isolate the square root**:
   \[
   10 - 4 = 2\sqrt{3v + 3},
   \]
   which simplifies to:
   \[
   6 = 2\sqrt{3v + 3}.
   \]

3. **Divide both sides by 2**:
   \[
   3 = \sqrt{3v + 3}.
   \]

4. **Square both sides again**:
   \[
   9 = 3v + 3.
   \]

5. **Solve for \(v\)**:
   \[
   3v = 9 - 3,
   \]
   \[
   3v = 6 \quad \Rightarrow \quad v = 2.
   \]

6. **Check the solution** in the original equation:
   - Left side:
     \[
     \sqrt{3(2) + 10} = \sqrt{6 + 10} = \sqrt{16} = 4.
     \]
   - Right side:
     \[
     \sqrt{3(2) + 3} + 1 = \sqrt{6 + 3} + 1 = \sqrt{9} + 1 = 3 + 1 = 4.
     \]

Since both sides equal 4, the solution \(v = 2\) is verified.

### Summary
The solution to the equation is \(v = 2\).

Would you like more details or do you have any questions? Here are some follow-up topics you might find interesting:

1. What are the steps for solving other types of radical equations?
2. How can we verify solutions for equations involving square roots?
3. What are common mistakes when working with square roots?
4. How does squaring both sides of an equation affect the solution?
5. What is the significance of extraneous solutions in equations? 

**Tip**: Always check your solutions, especially when squaring both sides, as this can introduce extraneous solutions.

Solve the equation \(\sqrt{3v + 10} = \sqrt{3v + 3} + 1\).

Discover how to solve the radical equation \(\sqrt{3v + 10} = \sqrt{3v + 3} + 1\) step-by-step. This guide covers essential algebraic concepts and properties of square roots, ideal for students in Grades 9-12.

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