1. **State the problem:** We need to divide the polynomial $5x^3 + 8x^2 - x + 6$ by the binomial $x + 2$. 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 numerator $5x^3$ by the leading term of the denominator $x$ to get $5x^2$. 4. **Multiply and subtract:** Multiply $5x^2$ by $x + 2$ to get $5x^3 + 10x^2$. Subtract this from the original polynomial: $$ (5x^3 + 8x^2 - x + 6) - (5x^3 + 10x^2) = 8x^2 - 10x^2 - x + 6 = -2x^2 - x + 6 $$ 5. **Repeat the process:** Divide the new leading term $-2x^2$ by $x$ to get $-2x$. 6. **Multiply and subtract:** Multiply $-2x$ by $x + 2$ to get $-2x^2 - 4x$. Subtract: $$ (-2x^2 - x + 6) - (-2x^2 - 4x) = -x + 4x + 6 = 3x + 6 $$ 7. **Repeat again:** Divide $3x$ by $x$ to get $3$. 8. **Multiply and subtract:** Multiply $3$ by $x + 2$ to get $3x + 6$. Subtract: $$ (3x + 6) - (3x + 6) = 0 $$ 9. **Conclusion:** The quotient is $5x^2 - 2x + 3$ and the remainder is $0$. **Final answer:** $$\frac{5x^3 + 8x^2 - x + 6}{x + 2} = 5x^2 - 2x + 3$$