The distribution should actually result in \log^{2}2+2\left(\log2\right)\left(\log5\right)+\log^{2}5+2\left(\log2\right)\left(\log5\right)-2\left(\left(\log^{2}2\right)\left(\log5\right)+\left(\log2\right)\left(\log^{2}5\right)\right)

Let's correctly distribute and simplify the expression step by step based on the correct distribution.

### Step 1: Reevaluate the original expression.
We have the expression:
\[
\left(\log 2\right)\left(\log 50\right) + \left(\log 5\right)\left(\log 20\right) - \left(\log 100\right)\left(\log 5\right)\left(\log 2\right)
\]

### Step 2: Rewrite the logarithms.
We express all logarithms in terms of \(\log 2\) and \(\log 5\):
\[
\log 50 = \log (2 \times 5^2) = \log 2 + 2\log 5
\]
\[
\log 20 = \log (2^2 \times 5) = 2\log 2 + \log 5
\]
\[
\log 100 = 2\log (10) = 2(\log 2 + \log 5)
\]

### Step 3: Substitute and distribute the terms.
Substitute these expressions back into the original expression:

1. \(\left(\log 2\right)\left(\log 50\right)\):
\[
\log 2 \times (\log 2 + 2\log 5) = \log^2 2 + 2\log 2 \times \log 5
\]

2. \(\left(\log 5\right)\left(\log 20\right)\):
\[
\log 5 \times (2\log 2 + \log 5) = 2\log 5 \times \log 2 + \log^2 5
\]

3. \(\left(\log 100\right)\left(\log 5\right)\left(\log 2\right)\):
\[
2(\log 2 + \log 5) \times \log 5 \times \log 2
\]

### Step 4: Simplify the expression.
Now, substitute and combine the terms:
\[
(\log^2 2 + 2\log 2 \times \log 5) + (2\log 5 \times \log 2 + \log^2 5) - 2(\log 2 \times \log 5 + \log^2 2 \times \log 5 + \log 2 \times \log^2 5)
\]

Expand and combine:
\[
\log^2 2 + 2\log 2 \times \log 5 + 2\log 5 \times \log 2 + \log^2 5 - 2(\log^2 2 \times \log 5 + \log 2 \times \log^2 5 + \log 2 \times \log 5)
\]

### Step 5: Group and cancel like terms.
Let's group and simplify:
\[
(\log^2 2 + \log^2 5) + 4\log 2 \times \log 5 - 2\log^2 2 \times \log 5 - 2\log 2 \times \log^2 5 - 2\log 2 \times \log 5
\]

Now simplify further:
\[
\log^2 2 + \log^2 5 + 2\log 2 \times \log 5 - 2(\log^2 2 \times \log 5 + \log 2 \times \log^2 5)
\]

### Final Simplified Expression:
This final expression cannot be simplified further directly without numerical values, so the final simplified result is:
\[
\boxed{\log^2 2 + \log^2 5 + 2\log 2 \times \log 5 - 2(\log^2 2 \times \log 5 + \log 2 \times \log^2 5)}
\]

Would you like more details or have any other questions?

### Related Questions:
1. How can you evaluate \(\log^2 3 + 2\log 3 \times \log 5 - 2(\log^2 3 \times \log 5 + \log 3 \times \log^2 5)\) without a calculator?
2. What is the significance of grouping logarithmic terms in simplifications?
3. Can we factorize the final expression to get a more compact form?
4. How does this method compare to direct substitution of numerical values for \(\log 2\) and \(\log 5\)?
5. How would the expression change if we were dealing with natural logarithms instead of common logarithms?

### Tip:
When simplifying logarithmic expressions, carefully track the distribution of terms and apply logarithm identities systematically to avoid errors.

Learn how to simplify the logarithmic expression \( \log^{2}2 + 2(\log2)(\log5) + \log^{2}5 + 2(\log2)(\log5) - 2((\log^{2}2)(\log5) + (\log2)(\log^{2}5)) \) step-by-step. This problem involves the distribution of logarithmic terms and systematic simplification, critical in advanced mathematics.

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