We are given two equations:

1. \((x + y)\left(4 + \frac{1}{xy}\right) = 1\)

2. \((4x + \frac{1}{x})(4y + \frac{1}{y}) = -20\)

Our goal is to solve this system of equations for \(x\) and \(y\).

First, let's simplify the first equation:

\[(x + y)\left(4 + \frac{1}{xy}\right) = 1\]

Expanding the terms, we have:

\[4(x + y) + \frac{x+y}{xy} = 1\]

\[4(x + y) + \left(\frac{1}{x} + \frac{1}{y}\right) = 1\]

Now let's simplify the second equation:

\[(4x + \frac{1}{x})(4y + \frac{1}{y}) = -20\]

Expanding this, we have:

\[16xy + 4y(\frac{1}{x}) + 4x(\frac{1}{y}) + \frac{1}{xy} = -20\]

\[16xy + \frac{4y}{x} + \frac{4x}{y} + \frac{1}{xy} = -20\]

Let's make a substitution to simplify the problem. If we let \(a = \frac{1}{x}\) and \(b = \frac{1}{y}\), then \(x = \frac{1}{a}\) and \(y = \frac{1}{b}\).

Now, substitute into the two equations:

**First equation:**

\[\left(\frac{1}{a} + \frac{1}{b}\right)\left(4 + ab\right) = 1\]

This simplifies to:

\[\left(\frac{a + b}{ab}\right)(4 + ab) = 1\]

Multiply both sides by \(ab\):

\[(a + b)(4 + ab) = ab\]

\[4a + 4b + a^2b + ab^2 = ab\]

\[a^2b + ab^2 + 4a + 4b - ab = 0\tag{3}\]

**Second equation:**

\[\left(4\left(\frac{1}{a}\right) + a\right)\left(4\left(\frac{1}{b}\right) + b\right) = -20\]

This simplifies to:

\[\left(\frac{4}{a} + a\right)\left(\frac{4}{b} + b\right) = -20\]

Multiply both factors:

\[\left(\frac{4 + a^2}{a}\right)\left(\frac{4 + b^2}{b}\right) = -20\]

Multiply both sides by \(ab\):

\[(4 + a^2)(4 + b^2) = -20ab\]

Again, we expand and simplify:

\[
(4 + a^2)(4 + b^2) = 16 + 4b^2 + 4a^2 + a^2b^2 = -20ab
\]

Combining terms gives:

\[
a^2b^2 + 4a^2 + 4b^2 + 16 + 20ab = 0\tag{4}
\]

We solve equations (3) and (4) simultaneously.

**From equation (3):**

\[a^2b + ab^2 - ab = -4(a + b)\]

\[ab(a + b - 1) = -4(a + b)\]

Assuming \(a+b \neq 0\), then:

\[ab(a + b - 1) = -4(a + b)\]

Dividing by \(a+b\):

\[ab = -4 \quad \text{if} \quad a + b - 1 \neq 0 \]

Substituting \(ab = -4\) into equation (4):

\[16 + 4a^2 + 4b^2 + a^2b^2 - 20ab = 0\]

Given \(ab = -4\), then \(ab^2 = -4b\) and \(a^2b = -4a\):

\[a^2(-4) + (-4)b^2 - 16 + 20(ab) = 0\]

\[4a^2 + 4b^2 + 16 + 20(-4) = 0\]

This simplifies further into a contradiction. Therefore, check mathematical operations in equations \(a, b\). Note the mathematical error leads to no viable solutions under these assumptions. Further numerical algorithms would be beneficial to validate a solution.

For ease, solve a combination \([A = x+y], [B = xy]\) with current solutions for simplification such as \(s = a+b, p = ab.\)