If x + 1/x = 5 , then the value of x^4 + 3x^3 + 5x^2 + 3x + 1x^4 + 1 = ?

Answer: Option A

$$\begin{array}{rl}& x+\frac{1}{x}=5\\ & \left(\text{By squaring both sides}\right)\\ & x^2+\frac{1}{x^2}+2.x.\frac{1}{x}={\left(5\right)}^2\\ & x^2+\frac{1}{x^2}=23\\ & \text{Now,}\\ & \frac{x^4+3x^3+5x^2+3x+1}{x^4+1}\\ & \text{Divided by }{x}^2,\\ & \Rightarrow \frac{\frac{x^4}{x^2}+\frac{3x^3}{x^2}+\frac{5x^2}{x^2}+\frac{3x}{x^2}+\frac{1}{x^2}}{\frac{x^4}{x^2}+\frac{1}{x^2}}\\ & \Rightarrow \frac{x^2+3x+5+\frac{3}{x}+\frac{1}{x^2}}{x^2+\frac{1}{x^2}}\\ & \Rightarrow \frac{x^2+\frac{1}{x^2}+3\left(x+\frac{1}{x}\right)+5}{x^2+\frac{1}{x^2}}\\ & \Rightarrow \frac{23+3\left(5\right)+5}{23}\\ & \Rightarrow \frac{43}{23}\end{array}$$