(x²-y-1) (y³ - 1)

To simplify the expression \((x^2 - y - 1)(y^3 - 1)(x^2 - y - 1)(y^3 - 1) \div ((x + y)(2x + y^2))\), we can start by breaking it down step by step.

First, let's observe the structure of the expression:
- The numerator is \((x^2 - y - 1)^2 (y^3 - 1)^2\) because we have the same terms appearing twice each.
- The denominator is \((x + y)(2x + y^2)\).

Now, we can write the full expression as follows:

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

Next, let's investigate the components:
1. **Factor \(y^3 - 1\)**: This can be factored using the difference of cubes:
\[
y^3 - 1 = (y - 1)(y^2 + y + 1).
\]

2. Therefore,
\[
(y^3 - 1)^2 = ((y - 1)(y^2 + y + 1))^2 = (y - 1)^2(y^2 + y + 1)^2.
\]

So the numerator becomes:
\[
(x^2 - y - 1)^2 (y - 1)^2 (y^2 + y + 1)^2.
\]

3. The denominator, \((x + y)(2x + y^2)\), remains as is.

So we have:
\[
\frac{(x^2 - y - 1)^2 (y - 1)^2 (y^2 + y + 1)^2}{(x + y)(2x + y^2)}.
\]

At this stage, the simplification largely depends on further factorization or substitution possibilities, if any, involving \(x\), \(y\), and specific cases.

However, unless we have specific values for \(x\) and \(y\) to further influence the simplification or specific requirements, this is a relatively condensed form of our expression.

In summary, the simplified expression is:
\[
\frac{(x^2 - y - 1)^2 (y - 1)^2 (y^2 + y + 1)^2}{(x + y)(2x + y^2)}.
\]