To solve the equation

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

we need to simplify and combine like terms on both sides of the equation. Let's start by rewriting the equation:

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

First, combine the constant terms on the left side:

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

Next, let's move all terms to one side of the equation to set it equal to zero:

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

Combine like terms:

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

Now, let's find a common denominator for the terms involving \(x\):

\[
2x^5 - x^4 - 3x^3 - x^2 - 4x - 5 = 0.
\]

This is a polynomial equation in \(x\). Solving this polynomial equation analytically can be quite complex, so we may need to use numerical methods or graphing techniques to find the roots.

However, we can check for any obvious rational roots using the Rational Root Theorem, which states that any rational root, in the form of \(\frac{p}{q}\), must be a factor of the constant term divided by a factor of the leading coefficient. Here, the constant term is \(-5\) and the leading coefficient is \(2\).

Possible rational roots are \(\pm 1, \pm 5, \pm \frac{1}{2}, \pm \frac{5}{2}\).

Testing these values:

1. \(x = 1\):
\[
2(1)^5 - (1)^4 - 3(1)^3 - (1)^2 - 4(1) - 5 = 2 - 1 - 3 - 1 - 4 - 5 = -12 \neq 0.
\]

2. \(x = -1\):
\[
2(-1)^5 - (-1)^4 - 3(-1)^3 - (-1)^2 - 4(-1) - 5 = -2 - 1 + 3 - 1 + 4 - 5 = -2 \neq 0.
\]

3. \(x = \frac{1}{2}\):
\[
2\left(\frac{1}{2}\right)^5 - \left(\frac{1}{2}\right)^4 - 3\left(\frac{1}{2}\right)^3 - \left(\frac{1}{2}\right)^2 - 4\left(\frac{1}{2}\right) - 5 = \frac{2}{32} - \frac{1}{16} - \frac{3}{8} - \frac{1}{4} - 2 - 5 = \frac{1}{16} - \frac{1}{16} - \frac{6}{16} - \frac{4}{16} - 2 - 5 = -7 \neq 0.
\]

4. \(x = \frac{-1}{2}\):
\[
2\left(\frac{-1}{2}\right)^5 - \left(\frac{-1}{2}\right)^4 - 3\left(\frac{-1}{2}\right)^3 - \left(\frac{-1}{2}\right)^2 - 4\left(\frac{-1}{2}\right) - 5 = -\frac{2}{32} - \frac{1}{16} + \frac{3}{8} - \frac{1}{4} + 2 - 5 = -\frac{1}{16} - \frac{1}{16} + \frac{6}{16} - \frac{4}{16} + 2 - 5 = -3 \neq 0.
\]

Since none of these rational roots satisfy the equation, we conclude that the roots are either irrational or complex. Numerical methods or graphing tools would be necessary to find the approximate solutions to this polynomial equation.