$\text{ Let }f:R\to R\text{ such that }{f}^{\text{'}\text{'}}\left(x\right)-2f^{\text{'}}\left(x\right)+f\left(x\right)=2e^x\text{ and }{f}^{\text{'}}\left(x\right)>0\forall x\in R\text{ then }$which of the following can be correct

$\frac{d}{dx}\left(e^{-x}\left(f^{\text{'}}\left(x\right)-f\left(x\right)\right)\right)=2$

$\Rightarrow e^{-x}\left(f^{\text{'}}\left(x\right)-f\left(x\right)\right)=2x+c_1$

$\Rightarrow f\left(x\right)=\left(x^2+c_{1}n+c_2\right)e^x,f^{\text{'}}\left(x\right)=e^x\left(x^2+\left(c_1+2\right)x+c_1+c_2\right)$

$\text{ Gives }{f}^{\text{'}}\left(x\right)>0\Rightarrow x^2\left(c_1+2\right)x+c_1+c_2>0\Rightarrow \Delta <0$

$\Rightarrow {\left(C_1+2\right)}^2-4C_1-C_2<0$

$\Rightarrow C_1^2-4C_2+4<0$

$\Rightarrow C_1^2-4C_2<-4$

$\Rightarrow C_1^2-4C_2<0$

$\therefore f\left(x\right)>0\forall x\in R$

$\therefore \left|f\right(x\left)\right|=f\left(x\right)\forall x\in R \therefore f\left(x\right)>e^x$