(i) \( \int_{-4}^{-1} x^{2}(3-4 x) d x \)

Calculate the integral \( \int_{-4}^{-1} x^{2}(3-4x) dx \). Evaluate the integral by following steps: - step0: Evaluate using formulas and rules: \(\int_{-4}^{-1} x^{2}\left(3-4x\right) dx\) - step1: Multiply the terms: \(\int_{-4}^{-1} \left(3x^{2}-4x^{3}\right) dx\) - step2: Evaluate the integral: \(\int \left(3x^{2}-4x^{3}\right) dx\) - step3: Use properties of integrals: \(\int 3x^{2} dx+\int -4x^{3} dx\) - step4: Evaluate the integral: \(x^{3}+\int -4x^{3} dx\) - step5: Evaluate the integral: \(x^{3}-x^{4}\) - step6: Return the limits: \(\left(x^{3}-x^{4}\right)\bigg |_{-4}^{-1}\) - step7: Calculate the value: \(318\) The integral of \(x^{2}(3-4x)\) from -4 to -1 is 318.