1. **State the problem:** A rectangular lawn measures 8 meters by 4 meters and is surrounded by a flower bed of uniform width $x$ meters. The total area of the lawn plus the flower bed is 165 square meters. We need to find the width $x$ of the flower bed. 2. **Set up the equation:** The total length including the flower bed is $8 + 2x$ and the total width is $4 + 2x$ because the flower bed surrounds all sides. 3. **Write the area equation:** $$\text{Total area} = (8 + 2x)(4 + 2x) = 165$$ 4. **Expand the product:** $$ (8 + 2x)(4 + 2x) = 8 \times 4 + 8 \times 2x + 2x \times 4 + 2x \times 2x = 32 + 16x + 8x + 4x^2 $$ 5. **Simplify:** $$ 4x^2 + 24x + 32 = 165 $$ 6. **Bring all terms to one side:** $$ 4x^2 + 24x + 32 - 165 = 0 $$ $$ 4x^2 + 24x - 133 = 0 $$ 7. **Divide entire equation by 4 to simplify:** $$ \frac{\cancel{4}x^2}{\cancel{4}} + \frac{24x}{4} - \frac{133}{4} = 0 $$ $$ x^2 + 6x - \frac{133}{4} = 0 $$ 8. **Use quadratic formula:** For $ax^2 + bx + c = 0$, $$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $$ Here, $a=1$, $b=6$, $c=-\frac{133}{4}$. 9. **Calculate discriminant:** $$ b^2 - 4ac = 6^2 - 4 \times 1 \times \left(-\frac{133}{4}\right) = 36 + 133 = 169 $$ 10. **Calculate roots:** $$ x = \frac{-6 \pm \sqrt{169}}{2} = \frac{-6 \pm 13}{2} $$ 11. **Find possible values:** - $$ x = \frac{-6 + 13}{2} = \frac{7}{2} = 3.5 $$ - $$ x = \frac{-6 - 13}{2} = \frac{-19}{2} = -9.5 $$ 12. **Interpret solution:** Width cannot be negative, so $x = 3.5$ meters. **Final answer:** The width of the flower bed is **3.5 meters**.