If x + 1/x = 3 , where x 0 , then the value of x^4 + 3x^3 + 5x^2 + 3x + 1x^4 + 1 = ?

Answer: Option A

$$\begin{array}{rl}& x+\frac{1}{x}=3\\ & \Rightarrow x^2+1=3x.....(i)\\ & \Rightarrow {\left(x^2+1\right)}^2={\left(3x\right)}^2\\ & \Rightarrow x^4+1+2x^2=9x^2\\ & \Rightarrow x^4+1=7x^2.....(ii)\\ & \therefore \frac{x^4+3x^3+5x^2+3x+1}{x^4+1}\\ & \Rightarrow \frac{7x^2+3x^3+5x^2+3x}{x^4+1}\\ & \Rightarrow \frac{12x^2+3x^3+3x}{7x^2}\\ & \text{From equation (i)}\\ & \Rightarrow \frac{12x+3\left(x^2+1\right)}{7x}\\ & \Rightarrow \frac{12x+3\times 3x}{7x}\\ & \Rightarrow \frac{21x}{7x}\\ & \Rightarrow 3\end{array}$$