1. **State the problem:** We need to find expressions for the dimensions of a rectangular prism whose volume is given by the polynomial $$x^3y + 63y^2 - 7x^2 - 9xy^3$$. 2. **Recall the formula:** The volume of a rectangular prism is the product of its length, width, and height. So, we want to factor the volume expression into three factors representing these dimensions. 3. **Group terms to factor:** Group the polynomial into parts that can be factored: $$x^3y - 7x^2 - 9xy^3 + 63y^2$$ Group as: $$(x^3y - 7x^2) + (-9xy^3 + 63y^2)$$ 4. **Factor each group:** From the first group, factor out $x^2$: $$x^2(xy - 7)$$ From the second group, factor out $-9y^2$: $$-9y^2(xy - 7)$$ 5. **Factor out the common binomial:** $$x^2(xy - 7) - 9y^2(xy - 7) = (xy - 7)(x^2 - 9y^2)$$ 6. **Recognize difference of squares:** $$x^2 - 9y^2 = (x - 3y)(x + 3y)$$ 7. **Final factorization:** $$x^3y + 63y^2 - 7x^2 - 9xy^3 = (xy - 7)(x - 3y)(x + 3y)$$ 8. **Interpretation:** The dimensions of the prism can be expressed as: - Length: $xy - 7$ - Width: $x - 3y$ - Height: $x + 3y$ These three expressions multiply to give the volume polynomial. **Answer:** The dimensions are $xy - 7$, $x - 3y$, and $x + 3y$.