1. The problem asks to simplify or express the given algebraic expressions involving $f(x)$ and polynomials in $x$. 2. We are given four expressions: (a) $\frac{f(x)}{(x + 5)^2}$ (b) $\frac{x - 4}{f(x)}$ (c) $(x - 4)^2 f(x)$ (d) $\frac{f(x)}{(x - 4)(x + 5)^2}$ 3. Since $f(x)$ is a function and no explicit form is given, these expressions are already in simplest form unless $f(x)$ is specified. 4. Important rules: - Division by zero is undefined, so $x \neq -5$ for expressions with $(x+5)^2$ in the denominator. - Similarly, $x \neq 4$ for expressions with $(x-4)$ in the denominator. 5. Therefore, the simplified forms are: (a) $\frac{f(x)}{(x + 5)^2}$ with domain $x \neq -5$ (b) $\frac{x - 4}{f(x)}$ with domain $f(x) \neq 0$ (c) $(x - 4)^2 f(x)$ with no domain restriction from denominator (d) $\frac{f(x)}{(x - 4)(x + 5)^2}$ with domain $x \neq 4, x \neq -5$ 6. Without more information about $f(x)$, these are the expressions in their simplest forms respecting domain restrictions. Final answer: (a) $\frac{f(x)}{(x + 5)^2}$ (b) $\frac{x - 4}{f(x)}$ (c) $(x - 4)^2 f(x)$ (d) $\frac{f(x)}{(x - 4)(x + 5)^2}$