1. **State the problem:** We need to find the function $(f - g)(x)$ given $f(x) = -2x^2 - 7x$ and $g(x) = x^2 + 9x$. 2. **Recall the formula:** The difference of two functions is defined as: $$ (f - g)(x) = f(x) - g(x) $$ 3. **Substitute the given functions:** $$ (f - g)(x) = (-2x^2 - 7x) - (x^2 + 9x) $$ 4. **Distribute the minus sign:** $$ (f - g)(x) = -2x^2 - 7x - x^2 - 9x $$ 5. **Combine like terms:** $$ (f - g)(x) = (-2x^2 - x^2) + (-7x - 9x) $$ $$ (f - g)(x) = -3x^2 - 16x $$ 6. **Final answer:** $$(f - g)(x) = -3x^2 - 16x$$ This is the polynomial in simplest form representing the difference of the two functions.