To solve the equation

\[
-\frac{1}{x} - 1 + 2x^2 - \frac{5}{x^3 - 1} = \frac{4}{x^2 + x + 1},
\]

we need to find the values of \( x \) that satisfy this equation. Let's proceed step by step.

First, let's rewrite the equation for clarity:

\[
-\frac{1}{x} - 1 + 2x^2 - \frac{5}{x^3 - 1} = \frac{4}{x^2 + x + 1}.
\]

### Step 1: Simplify the Denominators

Notice that \( x^3 - 1 \) can be factored using the difference of cubes:

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

So, the term \(\frac{5}{x^3 - 1}\) can be rewritten as:

\[
\frac{5}{(x - 1)(x^2 + x + 1)}.
\]

### Step 2: Common Denominator

To combine the terms on the left-hand side, we need a common denominator. The common denominator for the terms involving \( x \) and \( x^3 - 1 \) is \( x(x^3 - 1) \):

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

### Step 3: Rewrite Each Term with the Common Denominator

Rewrite each term with the common denominator \( x(x - 1)(x^2 + x + 1) \):

1. \(-\frac{1}{x}\):

\[
-\frac{1}{x} = -\frac{(x - 1)(x^2 + x + 1)}{x(x - 1)(x^2 + x + 1)} = -\frac{x^3 - 1}{x(x - 1)(x^2 + x + 1)}.
\]

2. \(-1\):

\[
-1 = -\frac{x(x - 1)(x^2 + x + 1)}{x(x - 1)(x^2 + x + 1)} = -\frac{x(x - 1)(x^2 + x + 1)}{x(x - 1)(x^2 + x + 1)}.
\]

3. \(2x^2\):

\[
2x^2 = \frac{2x^2 \cdot x(x - 1)(x^2 + x + 1)}{x(x - 1)(x^2 + x + 1)} = \frac{2x^2 \cdot x(x - 1)(x^2 + x + 1)}{x(x - 1)(x^2 + x + 1)}.
\]

4. \(-\frac{5}{x^3 - 1}\):

\[
-\frac{5}{x^3 - 1} = -\frac{5}{(x - 1)(x^2 + x + 1)} = -\frac{5x}{x(x - 1)(x^2 + x + 1)}.
\]

### Step 4: Combine the Terms

Combine all the terms over the common denominator:

\[
-\frac{x^3 - 1}{x(x - 1)(x^2 + x + 1)} - \frac{x(x - 1)(x^2 + x + 1)}{x(x - 1)(x^2 + x + 1)} + \frac{2x^2 \cdot x(x - 1)(x^2 + x + 1)}{x(x - 1)(x^2 + x + 1)} - \frac{5x}{x(x - 1)(x^2 + x + 1)} = \frac{4}{x^2 + x + 1}.
\]

### Step 5: Simplify the Numerator

Combine the numerators:

\[
-\left(x^3 - 1\right) - x(x - 1)(x^2 + x + 1) + 2x^3(x - 1)(x^2 + x + 1) - 5x = 4.
\]

### Step 6: Solve the Equation

This equation is quite complex and may require numerical methods or further algebraic manipulation to solve. Alternatively, we can check for specific values of \( x \) that might satisfy the equation.

### Step 7: Check Specific Values

Let's check if \( x = 1 \) is a solution:

\[
-\frac{1}{1} - 1 + 2(1)^2 - \frac{5}{(1)^3 - 1} = \frac{4}{(1)^2 + 1 + 1}.
\]

Simplify:

\[
-1 - 1 + 2 - \frac{5}{0} = \frac{4}{3}.
\]

Since \(\frac{5}{0}\) is undefined, \( x = 1 \) is not a solution.

### Conclusion

The equation is complex and may not have a simple analytical solution. It might be best to use numerical methods or graphing techniques to find approximate solutions.