We are given the equation:

\[
(6x + 1)^2 + 3 = 10
\]

### Step 1: Isolate the squared term
First, subtract 3 from both sides to isolate the squared term:

\[
(6x + 1)^2 = 10 - 3
\]

\[
(6x + 1)^2 = 7
\]

### Step 2: Take the square root of both sides
Now take the square root of both sides:

\[
6x + 1 = \pm \sqrt{7}
\]

This gives us two possible equations:

\[
6x + 1 = \sqrt{7} \quad \text{or} \quad 6x + 1 = -\sqrt{7}
\]

### Step 3: Solve each equation for \(x\)

#### Case 1: \(6x + 1 = \sqrt{7}\)
Subtract 1 from both sides:

\[
6x = \sqrt{7} - 1
\]

Now divide by 6:

\[
x = \frac{\sqrt{7} - 1}{6}
\]

#### Case 2: \(6x + 1 = -\sqrt{7}\)
Subtract 1 from both sides:

\[
6x = -\sqrt{7} - 1
\]

Now divide by 6:

\[
x = \frac{-\sqrt{7} - 1}{6}
\]

### Final Answer:
The solutions to the equation are:

\[
x = \frac{\sqrt{7} - 1}{6} \quad \text{or} \quad x = \frac{-\sqrt{7} - 1}{6}
\]

Would you like further details or explanations on any step?

Here are 5 related questions:
1. How can you verify the solutions using substitution?
2. What is the discriminant and how does it relate to this solution?
3. How do you simplify expressions involving square roots?
4. Can you apply the quadratic formula to a similar equation?
5. What are other methods to solve quadratic equations besides factoring?

**Tip**: Always isolate the squared term first when solving equations of this form to simplify the process.

Learn how to solve the equation (6x + 1)^2 + 3 = 10 by isolating the squared term and applying the square root property. This step-by-step solution covers key algebraic methods, suitable for Grades 8-10 students.

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