1. Stating the problem: We need to divide the polynomial $2x^{3}+3x^{2}+5x-8$ by $x^{2}+2x+3$. 2. The formula used is polynomial long division, similar to numerical long division. 3. Divide the leading term of the dividend $2x^{3}$ by the leading term of the divisor $x^{2}$ to get the first term of the quotient: $$\frac{2x^{3}}{x^{2}}=2x$$ 4. Multiply the entire 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. Now divide the leading term of the new dividend $-x^{2}$ by the leading term of the divisor $x^{2}$: $$\frac{-x^{2}}{x^{2}} = -1$$ 7. Multiply the divisor by $-1$: $$-1(x^{2}+2x+3) = -x^{2} - 2x - 3$$ 8. Subtract this from the current dividend: $$(-x^{2} - x - 8) - (-x^{2} - 2x - 3) = \cancel{-x^{2}} - x - 8 - \cancel{-x^{2}} + 2x + 3 = (-x + 2x) + (-8 + 3) = x - 5$$ 9. Since the degree of the remainder $x - 5$ is less than the degree of the divisor $x^{2}+2x+3$, the division ends here. 10. Final answer: $$\frac{2x^{3}+3x^{2}+5x-8}{x^{2}+2x+3} = 2x - 1 + \frac{x - 5}{x^{2}+2x+3}$$