I want to find out the $\operatorname{span}(A)$, $A=(x^4+1, x^3-x^2-x, x^4+x-2, 1)$, by showing that $A$ is linearly dependent. To do this I basically tried to use Gaussian elimination to bring the matrix into the rref, as following:
$$A=(x^4+1, x^3-x^2-x, x^4+x-2, 1)$$ $$α_1(x^4+1)+α_2(x^3-x^2-x)+α_3(x^4+x-2)+α_4(x^2+1)=x^4(α_1+α_3)+x^3(α_2)+x^2(-α_2+α_4)+x(-α_2+α_3)+(-2α_3+α_1)$$
$P(x)=ax^4+bx^3+cx^2+dx^1+e$
$$a=α_1+α_3, b=α_2, c=-α_2+α_2, d=-α_2+α_3, e=-2α_3+α_1$$
Using this I built an augmented matrix which looks like:
$$\left[\begin{array}{rrrr|r} 1 & 0 & 1 & 0 & a \\ 0 & 1 & 0 & 0 & b \\ 0 & -1 & 0 & 1 & c \\ 0 & -1 & 1 & 0 & d \\ 1 & 0 & -2 & 0 & e \end{array}\right]$$
From here I did made this matrix into a Rref matrix:
$$\left[\begin{array}{rrrr|r} 1 & 0 & 0 & 0 & a-b-d\\ 0 & 1 & 0 & 0 & b \\ 0 & 0 & 1 & 0 & d+b \\ 0 & 0 & 0 & 1 & b+c \\ 0 & 0 & 0 & 0 & e-a+3d+3b \end{array}\right]$$
Therefore $α_1=a-b-d$, $α_2=b$, $α_3=b+b$, $α_4=b+c$, $α_5=e-a+3d+3b$ but I'm not sure how I can use this to show that $A$ is linearly dependent and then use that to find $\operatorname{span}(A).$