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The given equation is a cubic equation:

\[
x^3 - 8x^2 - 49x + 392 = 0
\]

Let's solve this step by step:

### Step 1: Try Factorization
We will check if we can find a simple rational root by using the Rational Root Theorem. The possible rational roots are the divisors of the constant term (392) divided by the divisors of the leading coefficient (1). The divisors of 392 are:

\[
\pm 1, \pm 2, \pm 4, \pm 7, \pm 14, \pm 28, \pm 49, \pm 56, \pm 98, \pm 196, \pm 392
\]

### Step 2: Synthetic Division or Substitution
By testing possible roots, we find that \( x = 7 \) is a root (substitute \( x = 7 \) into the equation and it equals 0).

### Step 3: Polynomial Division
Now, we can perform synthetic division of \( x^3 - 8x^2 - 49x + 392 \) by \( x - 7 \) to reduce the cubic equation to a quadratic:

\[
x^3 - 8x^2 - 49x + 392 = (x - 7)(x^2 - x - 56)
\]

### Step 4: Solve the Quadratic
Now we need to solve the quadratic equation:

\[
x^2 - x - 56 = 0
\]

Using the quadratic formula:

\[
x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-56)}}{2(1)}
\]
\[
x = \frac{1 \pm \sqrt{1 + 224}}{2} = \frac{1 \pm \sqrt{225}}{2}
\]
\[
x = \frac{1 \pm 15}{2}
\]

So, the two solutions are:

\[
x = \frac{1 + 15}{2} = 8, \quad x = \frac{1 - 15}{2} = -7
\]

### Step 5: List the Final Solution
The complete solution set for the cubic equation is:

\[
x = 7, x = 8, x = -7
\]

Thus, the solution set is:

\[
\boxed{7, 8, -7}
\]

Would you like more details on any of the steps or clarifications? Here are some related questions you might find useful:

1. How do you perform synthetic division for polynomials?
2. What is the Rational Root Theorem, and how does it work?
3. Can all cubic equations be factored similarly?
4. What are the rules for solving cubic equations in general?
5. How can we verify if a solution is correct for a polynomial?

**Tip:** When solving higher-degree polynomials, always try to find at least one rational root, which can help reduce the degree of the polynomial and simplify the solution process.

Find all solutions for the equation x^3 - 8x^2 - 49x + 392 = 0.

Learn how to solve the cubic equation x^3 - 8x^2 - 49x + 392 with a detailed step-by-step solution. This problem covers key algebraic concepts such as polynomial division, the Rational Root Theorem, and the quadratic formula. Suitable for students in Grades 10-12.

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