$(R/I)[x]=R[x]/I$
No. First off, $I$ is not an ideal of $R[X]$ (can you see why?)
What is true is that $$(R/I)[X]\simeq R[X]/I[X]$$ where $I[X]$ denotes the ideal (can you prove it is one whenever $I$ is an ideal in $R$?) in $R[X]$ made up of polynomials with coefficients in $I$.
Proof Define $\eta :R[X]\to (R/I)[X]$ by mapping the polynomial $r_0+r_1X+r_2X^2+\cdots$ to the polynomial $(r_0+I)+(r_1+I)X+\cdots$. Show it is an homomorphism and find the kernel.