= Solution
Use the graded <ring homomorphism>
$$
\varphi:k[x_0,x_1,x_2,x_3]\longrightarrow k[s,t],\qquad x_i\longmapsto s^{3-i}t^i,
$$
where source degree $m$ maps to target degree $3m$. The <homogeneous ideal> of the <twisted cubic> is $\ker\varphi$: a <homogeneous polynomial> vanishes on the parametrized curve precisely when its substitution vanishes identically. Here $k$ is infinite, being an <algebraically closed field>, so vanishing at all parameter points implies the polynomial identity.
Let $J$ be generated by
$$
q_1=x_0x_3-x_1x_2,\qquad q_2=x_0x_2-x_1^2,\qquad q_3=x_1x_3-x_2^2.
$$
Their substitutions vanish, hence $J\subseteq\ker\varphi$. To prove the reverse inclusion as an equality of ideals, we need more than a set-theoretic description of the zero locus.
Modulo $J$, replace $x_1^2$ by $x_0x_2$, $x_2^2$ by $x_1x_3$, and $x_1x_2$ by $x_0x_3$. Each replacement strictly decreases the sum of the exponents of $x_1$ and $x_2$, so the reduction terminates. It shows that the degree-$m$ quotient is spanned by the <standard monomials of the twisted cubic>
$$
x_0^ax_3^b\quad(a+b=m),\qquad
x_0^ax_1x_3^b\quad(a+b=m-1),\qquad
x_0^ax_2x_3^b\quad(a+b=m-1).
$$
For $m\ge1$ there are $(m+1)+m+m=3m+1$ of these. For $m=0$ the sole monomial is $1$. Their substituted images have $t$-exponents respectively $3b$, $3b+1$ and $3b+2$, so they are exactly the distinct <monomials>
$$
s^{3m-j}t^j,\qquad 0\le j\le3m.
$$
These are linearly independent. Consequently any degree-$m$ polynomial in $\ker\varphi$, after reduction modulo $J$, has all its reduced coefficients zero. It belongs to $J$. Applying this to each homogeneous component gives
$$
\boxed{I(X)=\ker\varphi=(x_0x_3-x_1x_2,\ x_0x_2-x_1^2,\ x_1x_3-x_2^2).}
$$
This also proves that the quotient embeds in the domain $k[s,t]$, so the displayed <ideal> is prime; no radicalization or saturation correction is concealed in the argument. Its image is the third <Veronese subring> $\bigoplus_{m\ge0}k[s,t]_{3m}$, and the calculation gives the <Hilbert polynomial> $3m+1$, consistent with degree three and genus zero.
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