## Elementary Linear Algebra 7th Edition

$S$ spans $P_3$.
Consider the combination $$a(x^2-2x)+b(x^3+8)+c(x^3-x^2)+d(x^2-4)=0.$$ Comparing the coefficients we get the system \begin{align*} b+c&=0\\ a+b-c+d&=0\\ -2a&=0\\ 8b-4d&=0. \end{align*} The coefficient matrix $$\left[ \begin{array} {cc}0&1&1&0\\1&1&1&1\\-2&0&0&0\\ 0&8&0&-4 \end{array} \right]$$ has zero determinant since $$\left| \begin{array} {cc}0&1&1&0\\1&1&1&1\\-2&0&0&0\\ 0&8&0&-4 \end{array} \right|=-2\left| \begin{array} {cc} 1&1&0\\ 1&1&1\\ 8&0&-4 \end{array} \right|=-2(-4-(-4))=0.$$ Hence, $S$ is linearly independent set of vectros and since $P_3$ has dimension $4$ then $S$ spans $P_3$.