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*}