Showing a basis for polynomials

Given the polynomials

P1 $(x)$ = $2 + x – x^2 – 2x^3$

P2 $(x)$ = $1 + x + 2x^2 + x^3$

P3 $(x)$ = $1 – x -x^2 + 2x^3$

P4 $(x)$ = $1 + x^2 + 2x^3$

Show that the set $S$ = (P1 $(x)$, P2 $(x)$, P3 $(x)$, P4 $(x)$) is a basis of $P^3$.

Now, I understand that to be a basis it has to be linearly independent and span $P^3$.

It spans $P^3$ as $P^3 = span(1, x, x^2, x^3)$ and each of the polynomials can be written in that form.

For linear independence I used the formula K1P1 + K2P2 + K3P3 + K4P4 = $0$ and found the result

$2$K1 + K2 + K3 + K4 = $0$

K1 + K2 – K3 = $0$

-K1 + $2$K2 – K3 + K4 = $0$

$-2$K1 + K2 + $2$K3 + $2$K4 = $0$

I know it can be solved directly or the determinant of the coefficient matrix can be used, however that gave a result of $-1$. I know you can use symbols like $r, s, t$ to solve directly but honestly I don’t understand how that works. Or have I done something wrong?