Let $f(X),g(X),h(X) \in \mathbb Z[X]$, we have, denoting by $p(X)_i$ the coefficient of $X^i$ of a polynomial $p(X) \in \mathbb Z[X]$
\begin{align*}
\bigl(20\cdot f(X) + 4X\cdot g(X) + X^3 \cdot h(X)\bigr)_0
&= 20 \cdot f(X)_0\\
\bigl(20 \cdot f(X) + 4X \cdot g(X) + X^3 \cdot h(X)\bigr)_1
&= 20 \cdot f(X)_1 + 4 \cdot g(X)_0\\
\bigl(20 \cdot f(X) + 4X \cdot g(X) + X^3 \cdot h(X)\bigr)_2
&= 20 \cdot f(X)_2 + 4 \cdot g(X)_1\\
\end{align*}
So, by definition of $T$, we have $20f(X) + 4Xg(X) + X^3h(X) \in T$, as $f,g,h$ were arbitrary, $(20, 4X, X^3) \subseteq T$.
On the other side, if $p(X) = \sum_{i=0}^n p_iX^i \in T$ is given, we know $20 \mid p_0$, $4\mid p_1$, $4 \mid p_2$. That is, there are $\alpha, \beta, \gamma \in \mathbb Z$ such that $p_0 = 20\alpha$, $p_1 = 4\beta$, $p_2 = 4\gamma$. But then
\begin{align*}
p(X) &= \sum_{i=0}^n p_i X^i\\
&= p_0 + p_1 X + p_2 X^2 + X^3 \cdot \sum_{i=0}^{n-3} p_{i+3}X^i \\
&= 20 \cdot \alpha + 4X \cdot (\beta +\gamma X)+ X^3 \cdot \sum_{i=0}^{n-3} p_{i+3}X^i\\
&\in (20, 4X, X^3)
\end{align*}