Let's solve the given mathematical expression step by step:

$\sqrt{8} - 2 \sin 30^\circ - |1 - \sqrt{2}| + \left(\frac{1}{2}\right)^{-2} - (\pi - 2020)^0$

1. Simplify each term:

   • $\sqrt{8}$: $\sqrt{8} = \sqrt{4 \times 2} = 2\sqrt{2}$
   • $\sin 30^\circ$: $\sin 30^\circ = \frac{1}{2}$
   • $2 \sin 30^\circ$: $2 \sin 30^\circ = 2 \times \frac{1}{2} = 1$
   • $|1 - \sqrt{2}|$: $\sqrt{2} \approx 1.414 \Rightarrow 1 - \sqrt{2} \approx 1 - 1.414 = -0.414 \Rightarrow |1 - \sqrt{2}| = 0.414$
   • $\left(\frac{1}{2}\right)^{-2}$: $\left(\frac{1}{2}\right)^{-2} = \left(\frac{2}{1}\right)^2 = 2^2 = 4$
   • $(\pi - 2020)^0$: $(\pi - 2020)^0 = 1$

2. Substitute back into the expression:

   $2\sqrt{2} - 1 - 0.414 + 4 - 1$

3. Combine and simplify:

   • Combine the constants: $-1 - 0.414 - 1 = -2.414$
   • Combine the entire expression: $2\sqrt{2} - 2.414 + 4$
   • Simplify further: $2\sqrt{2} + 1.586$

So, the simplified value of the given expression is: $2\sqrt{2} + 1.586 \approx 4.414$

Would you like details or have any questions?

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Tip: When solving complex expressions, break them down into simpler parts and solve each part step by step to avoid errors.