1. **Stating the problem:** We are given that in pattern $k$, the difference between the total number of squares and the number of shaded squares is 271. 2. **Expressing the difference mathematically:** The problem states this difference can be shown as: $$k^2 + 3k + 1 = 271$$ 3. **Solving the quadratic equation:** Rewrite the equation: $$k^2 + 3k + 1 = 271$$ Subtract 271 from both sides: $$k^2 + 3k + 1 - 271 = 0$$ $$k^2 + 3k - 270 = 0$$ 4. **Using the quadratic formula:** The quadratic formula is: $$k = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$ where $a=1$, $b=3$, and $c=-270$. Calculate the discriminant: $$\Delta = b^2 - 4ac = 3^2 - 4 \times 1 \times (-270) = 9 + 1080 = 1089$$ Calculate the square root: $$\sqrt{1089} = 33$$ 5. **Find the two possible values for $k$:** $$k = \frac{-3 \pm 33}{2}$$ Calculate each: - $$k = \frac{-3 + 33}{2} = \frac{30}{2} = 15$$ - $$k = \frac{-3 - 33}{2} = \frac{-36}{2} = -18$$ 6. **Interpret the solution:** Since $k$ represents a pattern number (likely positive), we discard the negative value. **Final answer:** $$k = 15$$