Find Abc 0Eb3C5

1. **State the problem:** We need to find constants $A$, $B$, and $C$ such that $$x^2 + 2x - 1 = A(x-1)(x+2) + B(x-1)(x+2) + C(x-1)^2(x+2).$$ 2. **Combine like terms on the right side:** Notice that $A$ and $B$ multiply the same factor $(x-1)(x+2)$, so we can write: $$x^2 + 2x - 1 = (A + B)(x-1)(x+2) + C(x-1)^2(x+2).$$ 3. **Expand the factors:** - Expand $(x-1)(x+2)$: $$ (x-1)(x+2) = x^2 + 2x - x - 2 = x^2 + x - 2. $$ - Expand $(x-1)^2$: $$ (x-1)^2 = x^2 - 2x + 1. $$ - Then expand $C(x-1)^2(x+2)$: $$ C(x^2 - 2x + 1)(x+2) = C(x^3 + 2x^2 - 2x^2 - 4x + x + 2) = C(x^3 - x - 2). $$ 4. **Rewrite the right side:** $$ (A+B)(x^2 + x - 2) + C(x^3 - x - 2). $$ 5. **Expand the right side fully:** $$ (A+B)x^2 + (A+B)x - 2(A+B) + Cx^3 - Cx - 2C. $$ 6. **Group terms by powers of $x$:** $$ Cx^3 + (A+B)x^2 + ((A+B) - C)x - 2(A+B) - 2C. $$ 7. **Match coefficients with the left side $x^2 + 2x - 1$:** - Coefficient of $x^3$: $C = 0$ (since left side has no $x^3$ term) - Coefficient of $x^2$: $A + B = 1$ - Coefficient of $x$: $(A + B) - C = 2$ - Constant term: $-2(A + B) - 2C = -1$ 8. **Use $C=0$ to simplify:** - From $C=0$, the $x$ coefficient equation becomes $A + B = 2$. - But from $x^2$ coefficient, $A + B = 1$. 9. **Contradiction found:** The system is inconsistent because $A + B$ cannot be both $1$ and $2$. 10. **Conclusion:** The given equation cannot hold for any constants $A$, $B$, and $C$ as stated because the right side simplifies incorrectly. Possibly the problem has a typo or needs correction.