1. **State the problem:** Expand and simplify the expression $$(A-2y)(7+2y)$$ and express it in the form $$49 + By^2$$. 2. **Recall the distributive property (FOIL method):** $$(a+b)(c+d) = ac + ad + bc + bd$$ 3. **Apply the distributive property:** $$(A-2y)(7+2y) = A \times 7 + A \times 2y - 2y \times 7 - 2y \times 2y$$ 4. **Calculate each term:** $$A \times 7 = 7A$$ $$A \times 2y = 2Ay$$ $$-2y \times 7 = -14y$$ $$-2y \times 2y = -4y^2$$ 5. **Combine all terms:** $$7A + 2Ay - 14y - 4y^2$$ 6. **Compare with the given form:** The problem states the expression should be in the form $$49 + By^2$$. 7. **Analyze the constant term:** The constant term in our expansion is $$7A$$, which should equal $$49$$. 8. **Solve for A:** $$7A = 49 \implies A = \frac{49}{7} = 7$$ 9. **Substitute $$A=7$$ back into the expression:** $$7 \times 7 + 2 \times 7 \times y - 14y - 4y^2 = 49 + 14y - 14y - 4y^2$$ 10. **Simplify the $$y$$ terms:** $$14y - 14y = 0$$ 11. **Final simplified expression:** $$49 - 4y^2$$ 12. **Identify coefficient $$B$$:** The expression is $$49 + By^2$$, so $$B = -4$$. **Answer:** $$B = -4$$