1. **State the problem:** We need to divide the polynomial $2x^3 + 3x - 5$ by the binomial $x - 1$. 2. **Formula and method:** Polynomial division can be done using long division or synthetic division. Here, we use long division. 3. **Set up the division:** Divide the leading term of the dividend $2x^3$ by the leading term of the divisor $x$ to get $2x^2$. 4. **Multiply and subtract:** Multiply $2x^2$ by $x - 1$ to get $2x^3 - 2x^2$. Subtract this from the original polynomial: $$ (2x^3 + 0x^2 + 3x - 5) - (2x^3 - 2x^2) = 2x^2 + 3x - 5 $$ 5. **Repeat the process:** Divide $2x^2$ by $x$ to get $2x$. Multiply $2x$ by $x - 1$ to get $2x^2 - 2x$. Subtract: $$ (2x^2 + 3x - 5) - (2x^2 - 2x) = 5x - 5 $$ 6. **Continue:** Divide $5x$ by $x$ to get $5$. Multiply $5$ by $x - 1$ to get $5x - 5$. Subtract: $$ (5x - 5) - (5x - 5) = 0 $$ 7. **Conclusion:** The quotient is $2x^2 + 2x + 5$ and the remainder is $0$. **Final answer:** $$\frac{2x^3 + 3x - 5}{x - 1} = 2x^2 + 2x + 5$$