$f:R\to R f\left(x^2+x+3\right)+2f\left(x^2-3x+5\right)=6x^2-10x+17 \mathrm{\forall }x\in R\text{ then the value of }f\left(100\right)\text{ is }$

$\text{ obviously }f\text{ is a linear polynomial (Since degree of LHS and RHS is same)}$

$\begin{array}{l}\mathrm{let}f\left(x\right)=ax+b\text{ hence }f\left(x^2+x+3\right)+2f\left(x^2-3x+5\right)\\ \equiv 6x^2-10x+17\\ \Rightarrow \left[a\left(x^2+x+3\right)+b\right]+2\left[a\left(x^2-3x+5\right)+b\right]\\ \equiv 6x^2-10x+17\\ comparing coeff of x^2\text{ and coeff. of }x\\ \Rightarrow a+2a=6......\left(1\right)\\ \Rightarrow a-6a=-10......\left(2\right) a=2\\ comparing cons\tan ts\\ 3a+b+10a+2b=17\\ \Rightarrow 6+b+20+2b=17\\ \Rightarrow b=-3\\ \therefore f\left(x\right)=2x-3\\ \Rightarrow f\left(100\right)=197\end{array}$