To find the product function $(f \cdot g)(x)$ for $f(x) = x - 7$ and $g(x) = x^2 + 8x + 4$, use the formula $(f \cdot g)(x) = f(x) \cdot g(x)$ and multiply the polynomials: $(x - 7)(x^2 + 8x + 4)$. Distributing the terms produces $x^3 + 8x^2 + 4x - 7x^2 - 56x - 28$. Combining like terms simplifies the product to $(f \cdot g)(x) = x^3 + x^2 - 52x - 28$.