To calculate \( A = \sqrt[3]{\sqrt{5}} + 2 - \sqrt[3]{\sqrt{5} - 2} \), we can let:

\[
x = \sqrt[3]{\sqrt{5}}
\]

This gives us:

\[
A = x + 2 - \sqrt[3]{x^2 - 2}
\]

Next, we will simplify \( \sqrt[3]{x^2 - 2} \):

1. Calculate \( x^2 \):
\[
x^2 = \left( \sqrt[3]{\sqrt{5}} \right)^2 = \sqrt[3]{(\sqrt{5})^2} = \sqrt[3]{5}
\]

2. Substitute \( x^2 \) into the expression:
\[
A = x + 2 - \sqrt[3]{\sqrt{5} - 2}
\]

At this point, let's approximate the values:

- \( \sqrt{5} \approx 2.236 \)
- Therefore, \( \sqrt{5} - 2 \approx 0.236 \)

Now calculate:

\[
\sqrt[3]{0.236} \approx 0.620
\]
\[
x \approx \sqrt[3]{2.236} \approx 1.304
\]

Substituting these values back into the equation for \( A \):

\[
A \approx 1.304 + 2 - 0.620 = 2.684
\]

This can be further simplified or verified through algebraic manipulation or exact calculations if more precision is needed. Ultimately, for practical purposes, \( A \approx 2.684 \).