We are tasked with evaluating the limit:

$\lim_{x \to -2} \left( x^3 + 1 \right) \left( 1 - 2x \right)^2$

Step-by-Step Solution:

1. Substitute $x = -2$ directly into the expression:

   The expression is $\left( x^3 + 1 \right) \left( 1 - 2x \right)^2$. Let's substitute $x = -2$ into both parts:

   • For $x^3 + 1$: $(-2)^3 + 1 = -8 + 1 = -7$

   • For $(1 - 2x)^2$: $1 - 2(-2) = 1 + 4 = 5$ $(1 - 2x)^2 = 5^2 = 25$

2. Multiply the results:

   Now, multiply the results from the two parts: $(-7) \times 25 = -175$

Thus, the value of the limit is:

$\lim_{x \to -2} \left( x^3 + 1 \right) \left( 1 - 2x \right)^2 = -175$

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Would you like further clarification or details about any part of the process? Here are some additional questions that might interest you:

1. What happens if we try a similar limit as $x \to 0$ for this expression?
2. How can we simplify a more complicated limit involving products of functions?
3. What is the meaning of limits in the context of continuity?
4. What happens to the limit if we change $1 - 2x$ to $2 - x$?
5. How do limits apply to real-world situations, such as speed and acceleration?

Tip: When evaluating limits, especially with polynomials and powers, always check for direct substitution first. If that results in a well-defined value, you don't need any further steps!