1. **State the problem:** Find the product of the functions $f(x) = 4 - x^2$ and $g(x) = 2 - x$, i.e., find $(fg)(x) = f(x) \cdot g(x)$. 2. **Formula and rules:** The product of two functions is given by: $$ (fg)(x) = f(x) \times g(x) $$ We multiply the expressions for $f(x)$ and $g(x)$ and simplify. 3. **Intermediate work:** $$ (fg)(x) = (4 - x^2)(2 - x) $$ Use distributive property: $$ = 4 \times 2 - 4 \times x - x^2 \times 2 + x^2 \times x $$ $$ = 8 - 4x - 2x^2 + x^3 $$ Rearranged in standard polynomial form: $$ (fg)(x) = x^3 - 2x^2 - 4x + 8 $$ 4. **Answer:** The correct choice is (b) $x^3 - 2x^2 - 4x + 8$. --- 5. **Next problem:** Given $f(x) = 2x^2 + x - 3$ and $g(x) = x - 1$, express $$ f(x) \cdot g(x) - [f(x) + g(x)] $$ in standard form. 6. **Formula and rules:** First find $f(x) \cdot g(x)$, then subtract $f(x) + g(x)$. 7. **Intermediate work:** Calculate $f(x) \cdot g(x)$: $$ (2x^2 + x - 3)(x - 1) = 2x^2 \times x + 2x^2 \times (-1) + x \times x + x \times (-1) - 3 \times x - 3 \times (-1) $$ $$ = 2x^3 - 2x^2 + x^2 - x - 3x + 3 $$ Simplify: $$ = 2x^3 - 2x^2 + x^2 - x - 3x + 3 = 2x^3 - x^2 - 4x + 3 $$ Calculate $f(x) + g(x)$: $$ (2x^2 + x - 3) + (x - 1) = 2x^2 + x + x - 3 - 1 = 2x^2 + 2x - 4 $$ Subtract: $$ f(x) \cdot g(x) - [f(x) + g(x)] = (2x^3 - x^2 - 4x + 3) - (2x^2 + 2x - 4) $$ Distribute minus: $$ = 2x^3 - x^2 - 4x + 3 - 2x^2 - 2x + 4 $$ Combine like terms: $$ = 2x^3 - (x^2 + 2x^2) - (4x + 2x) + (3 + 4) $$ $$ = 2x^3 - 3x^2 - 6x + 7 $$ 8. **Final answer:** $$ 2x^3 - 3x^2 - 6x + 7 $$