Solve the equation `x(x+2)(x^2-1)=-1.`

Explanation

To solve the equation \( x(x+2)(x^2-1) = -1 \), we will follow these steps:

### Step 1: Rewrite the equation
We start with the equation:
\[ x(x+2)(x^2-1) = -1 \]
We can rewrite \( x^2 - 1 \) using the difference of squares:
\[ x^2 - 1 = (x+1)(x-1) \]
Thus, the equation becomes:
\[ x(x+2)(x+1)(x-1) = -1 \]

### Step 2: Move all terms to one side
To solve the equation, we can move -1 to the left side:
\[ x(x+2)(x+1)(x-1) + 1 = 0 \]

### Step 3: Expand the left side
Now we will expand the left side:
First, expand \( x(x+2) \):
\[ x(x+2) = x^2 + 2x \]
Next, expand \( (x+1)(x-1) \):
\[ (x+1)(x-1) = x^2 - 1 \]
Now, multiply these two results:
\[ (x^2 + 2x)(x^2 - 1) \]
Using the distributive property:
\[ = x^2(x^2 - 1) + 2x(x^2 - 1) = x^4 - x^2 + 2x^3 - 2x \]
Thus, we have:
\[ x^4 + 2x^3 - x^2 - 2x + 1 = 0 \]

### Step 4: Substitute \( y = x^2 + x \)
Let \( y = x^2 + x \). Then we can rewrite our equation:
\[ y^2 - 2y + 1 = 0 \]
This simplifies to:
\[ (y - 1)^2 = 0 \]
Thus, we find:
\[ y = 1 \]

### Step 5: Substitute back to find \( x \)
Now substituting back for \( y \):
\[ x^2 + x = 1 \]
Rearranging gives us:
\[ x^2 + x - 1 = 0 \]

### Step 6: Solve the quadratic equation
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 1, b = 1, c = -1 \):
\[ x = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1} \]
Calculating the discriminant:
\[ x = \frac{-1 \pm \sqrt{1 + 4}}{2} = \frac{-1 \pm \sqrt{5}}{2} \]
Thus, the solutions for \( x \) are:
\[ x = \frac{-1 + \sqrt{5}}{2} \quad \text{and} \quad x = \frac{-1 - \sqrt{5}}{2} \]

### Step 7: Final roots
Since the original equation is of degree 4, we also consider the roots from \( x(x + 2)(x^2 - 1) + 1 = 0 \). The roots are:
1. \( x = \frac{-1 + \sqrt{5}}{2} \)
2. \( x = \frac{-1 - \sqrt{5}}{2} \)
3. \( x = 0 \)
4. \( x = -2 \)

### Final Answer
The roots of the equation \( x(x+2)(x^2-1) = -1 \) are:
\[ x = \frac{-1 + \sqrt{5}}{2}, \quad x = \frac{-1 - \sqrt{5}}{2}, \quad x = 0, \quad x = -2 \]