Thank you for the clarification. The equation seems to involve a dot product or vector multiplication instead of the scalar equation I initially interpreted. However, the terms don't directly suggest vectors as we typically understand them in vector algebra. Let me adjust the interpretation:

The given expression can still be considered, but this time we're analyzing it with respect to the multiplication of expressions rather than vectors.

Given Expression:

$\left(\frac{-2}{x^2 - 4}\right) \cdot \left(\frac{x - 2}{4}\right) = 0$

Step 1: Analyze each term

• First Term: $\frac{-2}{x^2 - 4}$
• Second Term: $\frac{x - 2}{4}$

Step 2: Condition for the product to be zero

In mathematics, whether you're dealing with scalar products or vector dot products, if the product is zero, at least one of the factors must be zero.

First Term:

$\frac{-2}{x^2 - 4} = 0$ This cannot be zero because the numerator $-2$ is a non-zero constant.

Second Term:

$\frac{x - 2}{4} = 0$ For this term to be zero, $x - 2 = 0$, so $x = 2$.

Conclusion:

The expression equals zero when $x = 2$. The result is consistent whether considering the equation as a vector dot product scenario or a simple algebraic product of two terms.

Would you like more detailed steps or have any other questions?

Here are some related questions to explore:

1. How do you define a dot product for vector quantities?
2. What are the conditions under which a dot product equals zero?
3. How can you verify solutions in equations involving products?
4. What implications arise when the denominator in a fraction is zero?
5. How would this problem change if the numerator were zero instead of a constant?

Tip: When analyzing expressions involving multiplication, always consider the individual components. If any component equals zero, the entire expression is zero.