1. **State the problem:** Use the box method to distribute and simplify the product $$(-6x - 4)(-x^3 + 4x^2 - x + 6)$$. 2. **Recall the box method:** This method involves creating a grid where each term in the first polynomial multiplies each term in the second polynomial. Then, sum all the products. 3. **Set up the box:** - Rows: terms from $$-6x - 4$$ are $$-6x$$ and $$-4$$. - Columns: terms from $$-x^3 + 4x^2 - x + 6$$ are $$-x^3$$, $$4x^2$$, $$-x$$, and $$6$$. 4. **Multiply each pair:** - $$(-6x)(-x^3) = 6x^4$$ - $$(-6x)(4x^2) = -24x^3$$ - $$(-6x)(-x) = 6x^2$$ - $$(-6x)(6) = -36x$$ - $$(-4)(-x^3) = 4x^3$$ - $$(-4)(4x^2) = -16x^2$$ - $$(-4)(-x) = 4x$$ - $$(-4)(6) = -24$$ 5. **Write all terms:** $$6x^4, -24x^3, 6x^2, -36x, 4x^3, -16x^2, 4x, -24$$ 6. **Combine like terms:** - Combine $$-24x^3$$ and $$4x^3$$: $$-24x^3 + 4x^3 = -20x^3$$ - Combine $$6x^2$$ and $$-16x^2$$: $$6x^2 - 16x^2 = -10x^2$$ - Combine $$-36x$$ and $$4x$$: $$-36x + 4x = -32x$$ 7. **Final simplified expression:** $$6x^4 - 20x^3 - 10x^2 - 32x - 24$$ This is the product of the two polynomials using the box method.