## Solving the Polynomial Equation: (x²-2x-8)(x-1)²(x+5)⁶ = 0

This equation involves a product of multiple factors set equal to zero. To solve for the values of *x* that satisfy the equation, we can use the **Zero Product Property**: If the product of multiple factors is zero, at least one of the factors must be zero.

Let's break down each factor and find their roots:

### Factor 1: (x²-2x-8)

This is a quadratic expression. We can factor it as:

(x²-2x-8) = (x-4)(x+2)

Therefore, this factor equals zero when:

- x - 4 = 0 =>
**x = 4** - x + 2 = 0 =>
**x = -2**

### Factor 2: (x-1)²

This factor is a perfect square trinomial. It equals zero when:

- x - 1 = 0 =>
**x = 1**

Since the factor is squared, this solution has a **multiplicity of 2**.

### Factor 3: (x+5)⁶

This factor is also a power of a linear expression. It equals zero when:

- x + 5 = 0 =>
**x = -5**

This solution has a **multiplicity of 6**.

### Solutions to the Equation

Combining all the solutions from each factor, the complete set of solutions for the equation (x²-2x-8)(x-1)²(x+5)⁶ = 0 is:

**x = 4, x = -2, x = 1, x = -5**

It's important to remember that the solutions x = 1 and x = -5 are repeated solutions due to their multiplicities.

This means that the equation has a **total of 9 solutions** when considering the multiplicities:

- 4 distinct solutions (x = 4, x = -2, x = 1, x = -5)
- 2 solutions for x = 1
- 6 solutions for x = -5