1. **Problem statement:** We are given two functions $P(x)$ and $Q(x)$ with specified end behaviors: - $P(x) \to -\infty$ as $x \to +\infty$ and $P(x) \to +\infty$ as $x \to -\infty$. - $Q(x) \to -\infty$ as $x \to +\infty$ and $Q(x) \to -\infty$ as $x \to -\infty$. We want to describe the end behavior of the function $R(x) = P(x) \cdot Q(x)$. 2. **Recall the rule for end behavior of products:** The end behavior of a product of functions is the product of their end behaviors. That is, $$\lim_{x \to \pm \infty} R(x) = \lim_{x \to \pm \infty} P(x) \cdot \lim_{x \to \pm \infty} Q(x).$$ 3. **Analyze the behavior as $x \to +\infty$:** - $P(x) \to -\infty$ - $Q(x) \to -\infty$ Therefore, $$\lim_{x \to +\infty} R(x) = (-\infty) \times (-\infty) = +\infty.$$ 4. **Analyze the behavior as $x \to -\infty$:** - $P(x) \to +\infty$ - $Q(x) \to -\infty$ Therefore, $$\lim_{x \to -\infty} R(x) = (+\infty) \times (-\infty) = -\infty.$$ 5. **Conclusion:** - As $x \to +\infty$, $R(x) \to +\infty$. - As $x \to -\infty$, $R(x) \to -\infty$. This describes the end behavior of $R(x)$ based on the given behaviors of $P(x)$ and $Q(x)$.