The expression you've provided is:

\[
\lim_{{x \to 2}} \left( -3x^2 + 4x \right)
\]

Let's break it down and solve it step by step:

1. The limit notation means we want to evaluate the function \( -3x^2 + 4x \) as \( x \) approaches 2.
2. Since this is a polynomial, the limit can be evaluated directly by substituting \( x = 2 \) into the expression.

Now, let's substitute \( x = 2 \) into the function:

\[
-3(2)^2 + 4(2) = -3(4) + 8 = -12 + 8 = -4
\]

Thus, the limit is:

\[
\lim_{{x \to 2}} \left( -3x^2 + 4x \right) = -4
\]

Do you need further details or clarifications?

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Here are 5 related questions you might explore:
1. What is the limit of \( -3x^2 + 4x \) as \( x \to -2 \)?
2. How do you solve limits involving rational functions?
3. What is the difference between one-sided and two-sided limits?
4. Can you explain the limit laws that are applied when solving polynomial limits?
5. How do limits behave for functions with discontinuities?

**Tip:** For polynomial functions, limits can always be found by direct substitution if the function is continuous at the point of interest.

Learn how to evaluate the limit of the polynomial function -3x^2 + 4x as x approaches 2. This problem demonstrates the application of the direct substitution theorem, making it a great example for students in Grades 11-12.

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