1. **State the problem:** We need to graph the polar curve given by the equation $$r = -3 - 5 \cos \theta$$. 2. **Identify the type of curve:** This is a limaçon curve because it has the form $$r = a + b \cos \theta$$ where $$a = -3$$ and $$b = -5$$. 3. **Important rules:** - If $$|a| < |b|$$, the limaçon has an inner loop. - If $$|a| = |b|$$, the limaçon is a cardioid. - If $$|a| > |b|$$, the limaçon has no inner loop. Here, $$|a| = 3$$ and $$|b| = 5$$, so $$|a| < |b|$$, meaning the curve has an inner loop. 4. **Find key points:** - At $$\theta = 0$$, $$r = -3 - 5 \cos 0 = -3 - 5 = -8$$. - At $$\theta = \pi$$, $$r = -3 - 5 \cos \pi = -3 - 5(-1) = -3 + 5 = 2$$. - At $$\theta = \frac{\pi}{2}$$, $$r = -3 - 5 \cos \frac{\pi}{2} = -3 - 5(0) = -3$$. 5. **Plot points and shape:** - Negative $$r$$ values mean the point is plotted in the opposite direction of $$\theta$$. - The inner loop occurs because $$r$$ becomes negative for some $$\theta$$. 6. **Summary:** The curve is a limaçon with an inner loop, symmetric about the polar axis. **Final answer:** The polar curve $$r = -3 - 5 \cos \theta$$ is a limaçon with an inner loop.