1. **State the problem:** Divide the polynomial $3x^3 + 2x^2 - x + 4$ by the binomial $x - 1$. 2. **Formula and method:** We use polynomial long division or synthetic division to divide polynomials. The goal is to find quotient $Q(x)$ and remainder $R(x)$ such that: $$3x^3 + 2x^2 - x + 4 = (x - 1)Q(x) + R(x)$$ 3. **Perform polynomial long division:** - Divide the leading term $3x^3$ by $x$ to get $3x^2$. - Multiply $3x^2$ by $x - 1$ to get $3x^3 - 3x^2$. - Subtract this from the original polynomial: $$ (3x^3 + 2x^2) - (3x^3 - 3x^2) = 5x^2 $$ - Bring down the next term $-x$ to get $5x^2 - x$. - Divide $5x^2$ by $x$ to get $5x$. - Multiply $5x$ by $x - 1$ to get $5x^2 - 5x$. - Subtract: $$ (5x^2 - x) - (5x^2 - 5x) = 4x $$ - Bring down the last term $+4$ to get $4x + 4$. - Divide $4x$ by $x$ to get $4$. - Multiply $4$ by $x - 1$ to get $4x - 4$. - Subtract: $$ (4x + 4) - (4x - 4) = 8 $$ 4. **Result:** Quotient is $3x^2 + 5x + 4$ and remainder is $8$. 5. **Final answer:** $$\frac{3x^3 + 2x^2 - x + 4}{x - 1} = 3x^2 + 5x + 4 + \frac{8}{x - 1}$$