Do the polynomials x^3-2x^2+1,4x^2-x+3, and 3x-2 generate P_3(R)?Justify your answer.

Given polynomials are ${\displaystyle x^3-2x^2+1,4x^2-x+3}$ and 3x-2
It is required to find whether they generate ${\displaystyle P_3\left(R\right)}$.
Answer is No.
It is known that the vector space P3(R) consists of all polynomials having degree less than or equal to 3.
Let ${\displaystyle S=\{x^3-2x^2+1,4x^2-x+3\}}$ be a subset of ${\displaystyle P_3\left(R\right)}$
The dimension of ${\displaystyle P_3\left(R\right)}$ is 4. Every generating set must have at leas n elements.
The given polynomial set S has only three elements. So S cannot generate ${\displaystyle P_3\left(R\right)}$