Whether polynomials $(t-1)(t-2),(t-2)(t-3),(t-3)(t-4),(t-4)(t-6)$ are linearly independent. Question is to check if :
$(t-1)(t-2),(t-2)(t-3),(t-3)(t-4),(t-4)(t-6)\in \mathbb{R}[t]$ are linearly independent. 
Instead of writing linear combination and considering coefficient equations, I would like to say in the following way :
set of all polynomials of degree $\leq 2$ is a vector space with basis $1,t,t^2$ over $\mathbb{R}[t]$
 and there are 4 polnomials in the collection$(t-1)(t-2),(t-2)(t-3),(t-3)(t-4),(t-4)(t-6)\in \mathbb{R}[t]$
any collection of $n+1$ elements in a vector space of dimension $n$ is linearly dependent.
Thus, the collection $\{(t-1)(t-2),(t-2)(t-3),(t-3)(t-4),(t-4)(t-6)\}$ is linearly dependent in $\mathbb{R}[t]$
I would be thankful if some one can say if this justification is correct
i would be thankful if someone wants to say something more about this kind of checking.. 

Let $p_{i}(x)$ (where $1\leq i \leq n$) be $n$ polynomials in $\mathbb{R}[t]$. For checking of linearly dependently or independently these polynomials, in general we can using the following method. Let $d=Max_{1\leq i \leq n} deg(p_{i}(x))$ and $V$ be subspace of all polynomials of degree$\leq d$, then there exits subspace $W$ such that $$\mathbb{R}[t]=V\oplus W,$$
where also can prove that $W\cong \mathbb{R}[t]/V$ and $dim(\mathbb{R}[t])=dim(V)+dim(W)$, in which $dim(V)=d+1$.
Now we can say that the set $\{p_{i}(x) :  1\leq i \leq n \}$ with $n>d+1$ is linearly dependent set, in which $d=Max_{1\leq i \leq n} deg(p_{i}(x))$.
