1. **State the problem:** Find the product of $[(x - 7) + y][(x - 7) - y]$. 2. **Recall the formula:** This is a product of conjugates, which follows the difference of squares formula: $$ (A + B)(A - B) = A^2 - B^2 $$ where $A = (x - 7)$ and $B = y$. 3. **Apply the formula:** $$ [(x - 7) + y][(x - 7) - y] = (x - 7)^2 - y^2 $$ 4. **Expand $(x - 7)^2$:** $$ (x - 7)^2 = x^2 - 2 \times 7 \times x + 7^2 = x^2 - 14x + 49 $$ 5. **Write the final expression:** $$ x^2 - 14x + 49 - y^2 $$ 6. **Conclusion:** The product simplifies to $$ \boxed{x^2 - 14x + 49 - y^2} $$ This uses the difference of squares formula and expansion of a binomial squared.