1. **Stating the problem:** Solve the quadratic equation given the system of equations involving $p$ and $q$. 2. **Given equations:** $$3p + q = -5$$ $$p + q = -3$$ 3. **Step 1: Subtract the second equation from the first to eliminate $q$:** $$ (3p + q) - (p + q) = -5 - (-3) $$ $$ 3p + q - p - q = -5 + 3 $$ $$ 2p = -2 $$ 4. **Step 2: Solve for $p$:** $$ p = \frac{\cancel{2}p}{\cancel{2}} = \frac{-2}{2} = -1 $$ 5. **Step 3: Substitute $p = -1$ into the second equation to find $q$:** $$ -1 + q = -3 $$ $$ q = -3 + 1 = -2 $$ 6. **Step 4: Write the quadratic equation with found $p$ and $q$:** $$ x^2 + px + q = 0 $$ $$ x^2 - x - 2 = 0 $$ 7. **Step 5: Factor the quadratic:** $$ x^2 - x - 2 = (x - 2)(x + 1) = 0 $$ 8. **Step 6: Solve for $x$:** $$ x - 2 = 0 \Rightarrow x = 2 $$ $$ x + 1 = 0 \Rightarrow x = -1 $$ **Final answer:** $$ x = 2 \text{ or } x = -1 $$