1. **State the problem:** Factor the expression $ (2x-1)(4x^2-2x+1) $ completely. 2. **Recall the distributive property:** To factor or expand a product of two expressions, multiply each term in the first expression by each term in the second. 3. **Multiply the binomial and trinomial:** $$ (2x-1)(4x^2-2x+1) = 2x \cdot (4x^2-2x+1) - 1 \cdot (4x^2-2x+1) $$ 4. **Distribute each term:** $$ = 2x \cdot 4x^2 - 2x \cdot 2x + 2x \cdot 1 - 1 \cdot 4x^2 + 1 \cdot 2x - 1 \cdot 1 $$ $$ = 8x^3 - 4x^2 + 2x - 4x^2 + 2x - 1 $$ 5. **Combine like terms:** $$ 8x^3 - 4x^2 - 4x^2 + 2x + 2x - 1 = 8x^3 - 8x^2 + 4x - 1 $$ 6. **Check for further factoring:** The cubic polynomial $8x^3 - 8x^2 + 4x - 1$ does not factor nicely with simple methods, so this is the fully expanded form. **Final answer:** $$ (2x-1)(4x^2-2x+1) = 8x^3 - 8x^2 + 4x - 1 $$