A simple question confuses me a lot.

Q1
 $S=\{x\in\mathbb R^3 : 2x\_1+3x\_2^3-4x\_3^2=0\}$. Show that is not a subspace of $\mathbb R^3$ .

To show whether $S$ a subspace of $\mathbb R^3$ or not, we set $u,v\in S$.

$u=2u\_1+3u\_2^3-4u\_3^2, v=2v\_1+3v\_2^3-4v\_3^2$, and then follow the processes.

Q2 
 Let P denote the set of all real polynomials of degree less than or equal to 2.
 $S= {P \in P(R): P(x)=ax^2+bx+c} 

To show that, we set $u=a\_1x^2+b\_1x+c\_1$ and $v=a\_2x^2+b\_2x+c\_2$

Here is the question.. In Q1 we change the variable x into u and v. But in Q2 why we change the coefficients a,b,c into a1,a2,b1,b2,c1,c2 instead of changing the variable x???????