To solve the limit

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

we can evaluate the expression by direct substitution. Let's break it down:

1. First, substitute $x = 2$ into both parts of the expression:

   • For $x^2 - 3x$, when $x = 2$:

   $x^2 - 3x = 2^2 - 3(2) = 4 - 6 = -2.$

   • For $x + 1$, when $x = 2$:

   $x + 1 = 2 + 1 = 3.$

2. Now, multiply the two results:

   $(-2)(3) = -6.$

Thus, the limit is:

$\lim_{x \to 2} \left(x^2 - 3x\right)(x + 1) = -6.$

Would you like a more detailed explanation, or is there another concept you’d like to explore further?

Here are some related questions you might find useful:

1. What is the limit of $(x^2 - 4)(x + 1)$ as $x \to 2$?
2. How do we evaluate limits involving products of polynomials?
3. Can we apply L'Hopital's rule to this problem?
4. What happens if we substitute $x = 2$ in a rational expression like $\frac{x^2 - 3x}{x + 1}$?
5. How would the result change if we had a different value for $x$ instead of 2?

Tip: When evaluating limits of products, check if you can substitute directly. If both functions are continuous at the point, their product is also continuous, and you can just multiply their limits.