Values C D

1. **State the problem:** Given the equations $x^2 = b - ax$ and $x^3(x^3 + c) = d$, find which values of $c$ and $d$ satisfy these conditions. 2. **Rewrite the first equation:** From $x^2 = b - ax$, rearrange to get a quadratic in $x$: $$x^2 + ax - b = 0$$ 3. **Express $x^3$ in terms of $x$:** Multiply both sides of the first equation by $x$: $$x^3 = x(b - ax) = bx - a x^2$$ 4. **Substitute $x^2$ from the first equation into the expression for $x^3$:** Since $x^2 = b - ax$, then $$x^3 = bx - a(b - ax) = bx - ab + a^2 x = (b + a^2) x - ab$$ 5. **Calculate $x^3 + c$:** $$x^3 + c = (b + a^2) x - ab + c$$ 6. **Use the second equation $x^3(x^3 + c) = d$:** Substitute $x^3$ and $x^3 + c$: $$x^3(x^3 + c) = ((b + a^2) x - ab)((b + a^2) x - ab + c) = d$$ 7. **Expand the product:** $$((b + a^2) x - ab)((b + a^2) x - ab + c) = ((b + a^2) x - ab)^2 + c((b + a^2) x - ab)$$ 8. **Expand $((b + a^2) x - ab)^2$:** $$((b + a^2) x)^2 - 2 ab (b + a^2) x + (ab)^2 = (b + a^2)^2 x^2 - 2 a b (b + a^2) x + a^2 b^2$$ 9. **Substitute $x^2 = b - a x$ into the above:** $$(b + a^2)^2 (b - a x) - 2 a b (b + a^2) x + a^2 b^2 + c (b + a^2) x - a b c = d$$ 10. **Group terms by powers of $x$:** $$ (b + a^2)^2 b + a^2 b^2 - a (b + a^2)^2 x - 2 a b (b + a^2) x + c (b + a^2) x - a b c = d$$ 11. **Combine like terms for $x$:** Coefficient of $x$ is: $$-a (b + a^2)^2 - 2 a b (b + a^2) + c (b + a^2)$$ 12. **For the equation to hold for all $x$, the coefficient of $x$ must be zero:** $$-a (b + a^2)^2 - 2 a b (b + a^2) + c (b + a^2) = 0$$ 13. **Solve for $c$:** $$c (b + a^2) = a (b + a^2)^2 + 2 a b (b + a^2)$$ $$c = a (b + a^2) + 2 a b = a (b + a^2 + 2 b) = a (a^2 + 3 b) = a^3 + 3 a b$$ 14. **Calculate $d$ by substituting $c$ back and constant terms:** $$d = (b + a^2)^2 b + a^2 b^2 - a b c$$ 15. **Simplify $d$:** $$(b + a^2)^2 b + a^2 b^2 - a b (a^3 + 3 a b) = b (b^2 + 2 a^2 b + a^4) + a^2 b^2 - a^4 b - 3 a^2 b^2$$ $$= b^3 + 2 a^2 b^2 + a^4 b + a^2 b^2 - a^4 b - 3 a^2 b^2 = b^3$$ 16. **Final values:** $$c = a^3 + 3 a b, \quad d = b^3$$ **Answer:** The correct choice is $c = a^3 + 3 a b$ and $d = b^3$.