Find all values of a and b for which the vectors $f=-x^2-2x+5$, $g=ax+3$, and $h=x+b$ in $C^∞(R)$ are linearly independent
To solve this, from the equations I got the matrix $$ \begin{bmatrix} 1 & 0 & 0 \\ -2 & a & 1 \\ 5 & 3 & b \\ \end{bmatrix} $$
and then I found the determinant of the matrix and got $ a\cdot b-3=0 $ which would be $ a=3/b $ but since there is more variables than equations wouldn't it be inconsistent?