Given an ideal $I$ of $\mathbb{R}^5$ generated by two unknown polynomials. I know two homogenous polynomials $p_1 \in I$ and $p_2 \in I$ such that

1. $p_1$ is of degree 2 and up to a multiplicative constant the polynomial of smallest degree
2. $p_2$ is of degree 3 and up to a linear combination with $p_1$ the only polynomial of degree 3 in $I$.

Can I conclude that $p_1$ and $p_2$ generate $I$?

Given an ideal $I$ of $\mathbb{R}[X_1,X_2,X_3,X_4,X_5]$ generated by two unknown polynomials. I know two homogenous polynomials $p_1 \in I$ and $p_2 \in I$ such that

1. $p_1$ is of degree 2 and up to a multiplicative constant the polynomial of smallest degree
2. $p_2$ is of degree 3 and up to a linear combination with $p_1$ the only polynomial of degree 3 in $I$.

Can I conclude that $p_1$ and $p_2$ generate $I$?