1. **State the problem:** Simplify the expression $ (x+4y)(x-5y) $. 2. **Formula used:** Use the distributive property (FOIL method) for multiplying two binomials: $$ (a+b)(c+d) = ac + ad + bc + bd $$ 3. **Apply the formula:** $$ (x+4y)(x-5y) = x \cdot x + x \cdot (-5y) + 4y \cdot x + 4y \cdot (-5y) $$ 4. **Calculate each term:** $$ = x^2 - 5xy + 4xy - 20y^2 $$ 5. **Combine like terms:** $$ -5xy + 4xy = -1xy = -xy $$ 6. **Final simplified expression:** $$ x^2 - xy - 20y^2 $$ This is the product of the two binomials.