Describe the long run behavior of $$ f(x)=(x-4)^{2}(x+3)^{3}(x-3)^{2} $$ As $$ x \rightarrow-\infty, f(x) \rightarrow $$ ? As $$ x \rightarrow \infty, f(x) \rightarrow $$ ?

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1. Identify the degree of the polynomial:
   - The given function is 
     $$
     f(x)=(x-4)^{2}(x+3)^{3}(x-3)^{2}.
     $$
   - The exponents add up to the degree of the polynomial:
     $$
     \text{Degree} = 2 + 3 + 2 = 7.
     $$

2. Determine the leading coefficient:
   - When expanded, each factor contributes the term $$ x $$ multiplied by its exponent. The leading term is:
     $$
     x^2 \cdot x^3 \cdot x^2 = x^7.
     $$
   - The coefficient for $$ x^7 $$ is the product of the coefficients from each factor, each of which is $$ 1 $$. Thus, the leading coefficient is $$ 1 $$.

3. Analyze the end behavior based on the degree and leading coefficient:
   - Since the degree is odd ($$ 7 $$) and the leading coefficient is positive ($$ 1 $$):
     - As $$ x \rightarrow \infty $$, the term $$ x^7 $$ goes to $$ \infty $$.
     - As $$ x \rightarrow -\infty $$, the term $$ x^7 $$ goes to $$ -\infty $$.

4. Conclusion:
   - As $$ x \rightarrow -\infty $$, $$ f(x) \rightarrow -\infty $$.
   - As $$ x \rightarrow \infty $$, $$ f(x) \rightarrow \infty $$.
As $$ x $$ approaches negative infinity, $$ f(x) $$ approaches negative infinity. As $$ x $$ approaches positive infinity, $$ f(x) $$ approaches positive infinity.

Describe the long run behavior of \( f(x)=(x-4)^{2}(x+3)^{3}(x-3)^{2} \) As \( x \rightarrow-\infty, f(x) \rightarrow \) ? As \( x \rightarrow \infty, f(x) \rightarrow \) ?

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