1. **State the problem:** Divide the polynomial $2x^{3}+3x^{2}+5x-8$ by $x^{2}+2x+3$. 2. **Recall the division formula:** Polynomial division is similar to long division with numbers. We divide the highest degree term of the dividend by the highest degree term of the divisor, multiply the divisor by this result, subtract, and repeat. 3. **Divide the leading terms:** $\frac{2x^{3}}{x^{2}}=2x$. 4. **Multiply the divisor by $2x$:** $2x(x^{2}+2x+3)=2x^{3}+4x^{2}+6x$. 5. **Subtract this from the dividend:** $$\left(2x^{3}+3x^{2}+5x-8\right)-\left(2x^{3}+4x^{2}+6x\right) = \cancel{2x^{3}}+3x^{2}+5x-8 - \cancel{2x^{3}} -4x^{2} -6x = (3x^{2}-4x^{2}) + (5x-6x) -8 = -x^{2} - x -8$$ 6. **Repeat the division with the new polynomial:** Divide the leading term $-x^{2}$ by $x^{2}$ to get $-1$. 7. **Multiply the divisor by $-1$:** $-1(x^{2}+2x+3) = -x^{2} - 2x - 3$. 8. **Subtract this from the previous remainder:** $$(-x^{2} - x -8) - (-x^{2} - 2x - 3) = \cancel{-x^{2}} - x - 8 - \cancel{-x^{2}} + 2x + 3 = (-x + 2x) + (-8 + 3) = x - 5$$ 9. **Since the degree of the remainder $x - 5$ is less than the divisor degree 2, stop here.** 10. **Write the final answer:** $$\frac{2x^{3}+3x^{2}+5x-8}{x^{2}+2x+3} = 2x - 1 + \frac{x - 5}{x^{2}+2x+3}$$