1. **State the problem:** We have a cubic polynomial $$h x^3 + 7 x^2 - 48 x + 49$$ that leaves the same remainder when divided by $$x + k$$ and $$x - k$$. We need to find possible values of $$h$$ and $$k$$. 2. **Recall the Remainder Theorem:** The remainder when a polynomial $$f(x)$$ is divided by $$x - a$$ is $$f(a)$$. 3. **Apply the theorem:** The remainders when dividing by $$x + k$$ and $$x - k$$ are $$f(-k)$$ and $$f(k)$$ respectively. Since these remainders are equal, we have: $$f(-k) = f(k)$$ 4. **Write the polynomial:** $$f(x) = h x^3 + 7 x^2 - 48 x + 49$$ 5. **Calculate $$f(k)$$ and $$f(-k)$$:** $$f(k) = h k^3 + 7 k^2 - 48 k + 49$$ $$f(-k) = h (-k)^3 + 7 (-k)^2 - 48 (-k) + 49 = -h k^3 + 7 k^2 + 48 k + 49$$ 6. **Set the remainders equal:** $$h k^3 + 7 k^2 - 48 k + 49 = -h k^3 + 7 k^2 + 48 k + 49$$ 7. **Simplify the equation:** Subtract $$7 k^2 + 49$$ from both sides: $$h k^3 - 48 k = -h k^3 + 48 k$$ 8. **Bring all terms to one side:** $$h k^3 - 48 k + h k^3 - 48 k = 0$$ $$2 h k^3 - 96 k = 0$$ 9. **Factor out $$2 k$$:** $$2 k (h k^2 - 48) = 0$$ 10. **Solve for $$k$$ or $$h$$:** Either $$2 k = 0$$ which gives $$k = 0$$ (not interesting since division by $$x \\pm 0$$ is the same), or $$h k^2 - 48 = 0$$ which gives $$h k^2 = 48$$ 11. **Express $$h$$ in terms of $$k$$:** $$h = \frac{48}{k^2}$$ **Answer:** A possible set of values is any $$k \neq 0$$ and $$h = \frac{48}{k^2}$$. For example, if $$k = 1$$, then $$h = 48$$.