We are tasked with solving the equation:

\[
\left( x^2 - 2x - 10 \right)^{\frac{3}{2}} - 126 = -1
\]

### Step 1: Isolate the term with the exponent
We start by isolating the expression involving the exponent:

\[
\left( x^2 - 2x - 10 \right)^{\frac{3}{2}} = -1 + 126
\]
\[
\left( x^2 - 2x - 10 \right)^{\frac{3}{2}} = 125
\]

### Step 2: Eliminate the exponent
To eliminate the \(\frac{3}{2}\) exponent, raise both sides of the equation to the power of \(\frac{2}{3}\):

\[
x^2 - 2x - 10 = 125^{\frac{2}{3}}
\]

The cube root of 125 is 5, and squaring that gives:

\[
x^2 - 2x - 10 = 25
\]

### Step 3: Solve the quadratic equation
Now, solve the quadratic equation:

\[
x^2 - 2x - 10 = 25
\]

Move all terms to one side:

\[
x^2 - 2x - 35 = 0
\]

### Step 4: Factor the quadratic equation
We factor the quadratic equation:

\[
(x - 7)(x + 5) = 0
\]

Thus, the solutions are:

\[
x = 7 \quad \text{or} \quad x = -5
\]

### Step 5: Verify the solutions
To verify, substitute both \(x = 7\) and \(x = -5\) back into the original equation.

- For \(x = 7\):
  \[
  (7^2 - 2(7) - 10)^{\frac{3}{2}} - 126 = -1
  \]
  Simplifying: 
  \[
  (49 - 14 - 10)^{\frac{3}{2}} = 25^{\frac{3}{2}} = 125
  \]
  This holds true.

- For \(x = -5\):
  \[
  ((-5)^2 - 2(-5) - 10)^{\frac{3}{2}} - 126 = -1
  \]
  Simplifying:
  \[
  (25 + 10 - 10)^{\frac{3}{2}} = 25^{\frac{3}{2}} = 125
  \]
  This also holds true.

### Final Answer:
The solutions are:

\[
x = 7 \quad \text{or} \quad x = -5
\]

Would you like more details or have any questions?

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Here are 5 related questions:
1. How do you solve a quadratic equation by completing the square?
2. What are the properties of fractional exponents?
3. Can all quadratic equations be factored?
4. How can the quadratic formula be derived?
5. What is the geometric interpretation of quadratic equations?

**Tip:** Always check your solutions by substituting them back into the original equation to verify they are correct.

Explore a step-by-step solution for solving the equation (x^2 - 2x - 10)^(3/2) - 126 = -1. This detailed explanation covers key algebraic concepts, including quadratic equations and exponentiation rules, suitable for students in Grades 9-12.

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