1. State the problem: Expand and simplify the expression $(x-7)(x+3)$. 2. Formula and rule: Use the distributive property (FOIL) which expands products of binomials. $$ (a+b)(c+d) = ac + ad + bc + bd $$ 3. Apply FOIL to the given expression by multiplying each term in the first binomial by each term in the second. $$ (x-7)(x+3) = x(x+3) -7(x+3) $$ 4. Expand each product to show all terms. $$ = x^2 + 3x -7x -21 $$ 5. Combine like terms $3x -7x$ to get $-4x$ and then write the simplified polynomial. $$ = x^2 -4x -21 $$ 6. Final answer: The product simplifies to $x^2 -4x -21$.