1. **Stating the problem:** We are given the equation $$D(2x - 1) = A(x^2 - x + 1) + B(x - 2)$$ and need to find the values of constants $A$, $B$, and $D$ that satisfy this identity for all $x$. 2. **Understanding the problem:** This is a polynomial identity where the left side is a linear polynomial multiplied by $D$, and the right side is a sum of polynomials multiplied by constants $A$ and $B$. 3. **Rewrite the equation:** $$D(2x - 1) = A(x^2 - x + 1) + B(x - 2)$$ 4. **Expand the right side:** $$A(x^2 - x + 1) + B(x - 2) = A x^2 - A x + A + B x - 2 B$$ 5. **Group like terms:** $$= A x^2 + (-A + B) x + (A - 2 B)$$ 6. **Match coefficients with the left side:** The left side is $$D(2x - 1) = 2 D x - D$$ Since the left side is linear and the right side has a quadratic term $A x^2$, for the identity to hold for all $x$, the coefficient of $x^2$ must be zero: $$A = 0$$ 7. **Substitute $A=0$ back:** Right side becomes: $$0 imes x^2 + (-0 + B) x + (0 - 2 B) = B x - 2 B$$ 8. **Equate coefficients of $x$ and constant terms:** From $x$ terms: $$2 D = B$$ From constant terms: $$-D = -2 B$$ 9. **Solve the system:** From the constant terms: $$-D = -2 B \implies D = 2 B$$ From the $x$ terms: $$2 D = B$$ Substitute $D = 2 B$ into $2 D = B$: $$2 (2 B) = B \implies 4 B = B \implies 3 B = 0 \implies B = 0$$ Then: $$D = 2 B = 0$$ 10. **Final values:** $$A = 0, B = 0, D = 0$$ **Answer:** $$\boxed{A=0, B=0, D=0}$$