1. **State the problem:** We are given the volume function of a box as $$V(x) = 4x^3 - 130x^2 + 1000x$$ and need to show that when $x=1$, the volume is 874 cm³. 2. **Verify volume at $x=1$:** Substitute $x=1$ into the volume function: $$V(1) = 4(1)^3 - 130(1)^2 + 1000(1)$$ $$= 4 - 130 + 1000$$ $$= 874$$ This confirms the volume is 874 cm³ when $x=1$. 3. **Find other values of $x$ for volume 874:** Solve the equation: $$4x^3 - 130x^2 + 1000x = 874$$ Rewrite as: $$4x^3 - 130x^2 + 1000x - 874 = 0$$ 4. **Use the fact that $x=1$ is a root:** Since $x=1$ satisfies the equation, $(x-1)$ is a factor. Perform polynomial division or factor out $(x-1)$: Divide: $$4x^3 - 130x^2 + 1000x - 874 \div (x-1)$$ Using synthetic division: Coefficients: 4, -130, 1000, -874 Bring down 4. Multiply 4*1=4; add to -130: -126. Multiply -126*1=-126; add to 1000: 874. Multiply 874*1=874; add to -874: 0. So quotient is: $$4x^2 - 126x + 874$$ 5. **Solve quadratic:** $$4x^2 - 126x + 874 = 0$$ Use quadratic formula: $$x = \frac{126 \pm \sqrt{(-126)^2 - 4 \cdot 4 \cdot 874}}{2 \cdot 4}$$ Calculate discriminant: $$126^2 = 15876$$ $$4 \cdot 4 \cdot 874 = 13984$$ $$\sqrt{15876 - 13984} = \sqrt{1892}$$ 6. **Simplify and find roots:** $$x = \frac{126 \pm \sqrt{1892}}{8}$$ Approximate: $$\sqrt{1892} \approx 43.5$$ So, $$x_1 = \frac{126 + 43.5}{8} = \frac{169.5}{8} = 21.1875$$ $$x_2 = \frac{126 - 43.5}{8} = \frac{82.5}{8} = 10.3125$$ 7. **Final answer:** The values of $x$ that give a volume of 874 cm³ are: $$x = 1, 10.3125, 21.1875$$