Simplify x^2 + 2x + y^2x^3 - 5x^2 if x + y^2x = 5.

Answer: Option D

$$\begin{array}{rl}& \text{Given:}\\ & x+\frac{y^2}{x}=5\\ & \text{Put }y=2,x=1\\ & \text{Now,}\\ & \frac{x^2+2x+y^2}{x^3-5x^2}\\ & =\frac{{\left(1\right)}^2+2\times 1+{\left(2\right)}^2}{{\left(1\right)}^3-5{\left(1\right)}^2}\\ & =\frac{7}{-4}\\ & \text{Put }y=2\text{ in option and option D will satisfy the condition}\\ & \\ & \mathbf{Alternate}\mathbf{solution}:\\ & x+\frac{y^2}{x}=5\\ & x^2+y^2=5x\\ & x^2-5x=-y^2\\ & \text{Now, }\frac{x^2+2x+y^2}{x^3-5x^2}\\ & =\frac{5x+2x}{x\left(x^2-5x\right)}\\ & =\frac{7x}{x\left(-y^2\right)}\\ & =-\frac{7}{y^2}\end{array}$$