1. The problem is to expand and simplify the function $f(x) = (x^3 - 2x)(x^5 + 6x^2)$. 2. We use the distributive property (also called FOIL for binomials) to multiply each term in the first polynomial by each term in the second polynomial: $$f(x) = x^3 \cdot x^5 + x^3 \cdot 6x^2 - 2x \cdot x^5 - 2x \cdot 6x^2$$ 3. Multiply the powers of $x$ by adding exponents: - $x^3 \cdot x^5 = x^{3+5} = x^8$ - $x^3 \cdot 6x^2 = 6x^{3+2} = 6x^5$ - $-2x \cdot x^5 = -2x^{1+5} = -2x^6$ - $-2x \cdot 6x^2 = -12x^{1+2} = -12x^3$ 4. Write the expanded expression: $$f(x) = x^8 + 6x^5 - 2x^6 - 12x^3$$ 5. Rearrange terms in descending order of powers: $$f(x) = x^8 - 2x^6 + 6x^5 - 12x^3$$ 6. This is the simplified expanded form of the function. Final answer: $$\boxed{f(x) = x^8 - 2x^6 + 6x^5 - 12x^3}$$