1. **State the problem:** We want to express the rational function $$\frac{4x^3 - 5x^2 + 3x - 14}{x^2 + 2x - 1}$$ in the form $$Ax + B + \frac{Cx + D}{x^2 + 2x - 1}$$ and find constants $A$, $B$, $C$, and $D$. 2. **Formula and approach:** Since the degree of the numerator (3) is greater than the degree of the denominator (2), perform polynomial division: $$\frac{4x^3 - 5x^2 + 3x - 14}{x^2 + 2x - 1} = Ax + B + \frac{Cx + D}{x^2 + 2x - 1}$$ where $Ax + B$ is the quotient and $Cx + D$ is the remainder polynomial of degree less than 2. 3. **Perform polynomial division:** Divide $4x^3 - 5x^2 + 3x - 14$ by $x^2 + 2x - 1$. - First term: $4x^3 \div x^2 = 4x$, so $A = 4$. - Multiply divisor by $4x$: $$4x(x^2 + 2x - 1) = 4x^3 + 8x^2 - 4x$$ - Subtract: $$\left(4x^3 - 5x^2 + 3x - 14\right) - \left(4x^3 + 8x^2 - 4x\right) = -13x^2 + 7x - 14$$ - Next term: $-13x^2 \div x^2 = -13$, so $B = -13$. - Multiply divisor by $-13$: $$-13(x^2 + 2x - 1) = -13x^2 - 26x + 13$$ - Subtract: $$(-13x^2 + 7x - 14) - (-13x^2 - 26x + 13) = 33x - 27$$ 4. **Write the division result:** $$\frac{4x^3 - 5x^2 + 3x - 14}{x^2 + 2x - 1} = 4x - 13 + \frac{33x - 27}{x^2 + 2x - 1}$$ 5. **Identify constants:** $$A = 4, \quad B = -13, \quad C = 33, \quad D = -27$$ **Final answer:** $$\boxed{A=4, B=-13, C=33, D=-27}$$