To find the units digit of the expression $\((37)^X(21)^{X+1}(35)^{X+2}\)$, we only need to consider the units digit of each base number when raised to its respective power.

The units digit of the expression will be the units digit of the product of the units digits of each term:
Units digit of \([(units digit of 37$)^X \times(\text { units digit of } 21)^{X+1} \times(\text { units digit of } 35)^{X+2}$ ]\)
This simplifies to:
Units digit of $\(\left[(7)^X \times(1)^{X+1} \times(5)^{X+2}\right]\)$
Let's analyze the units digit for each part:
1. Units digit of $\((7)^X\)$ :

The units digits of powers of 7 follow a cycle:
- $\(7^1=7\)$
- $\(7^2=49 \Longrightarrow 9\)$
- $\(7^3=343 \Longrightarrow 3\)$
- $\(7^4=2401 \Longrightarrow 1\)$
- $\(7^5=16807 \Longrightarrow 7\)$

The cycle of units digits for powers of 7 is $(7,9,3,1)$, with a length of 4 . The units digit of $7^X$ can be $7,9,3$, or 1 depending on the value of $X$.
2. Units digit of $\((1)^{X+1}\)$ :

Any positive integer power of 1 has a units digit of 1 .
So, the units digit of $\((21)^{X+1}\)$ is always 1 .

3. Units digit of $\((5)^{X+2}\)$ :

Any positive integer power of 5 has a units digit of 5 .
- $\(5^1=5\)$
- $\(5^2=25 \Longrightarrow 5\)$

So, the units digit of $\((35)^{X+2}\)$ is always 5 .
Now, we need to find the units digit of the overall product:
Units digit of \([(Units digit of $7^X$ ) $\times 1 \times 5$ ]\)
Let's consider each possible units digit of $\(7^X\)$ and multiply by $\(1 \times 5=5\)$ :
- If the units digit of $\(7^X\)$ is 7 : Units digit of $\((7 \times 5)=$ Units digit of $(35)=\mathbf{5}\)$.
- If the units digit of $\(7^X\)$ is 9 : Units digit of $\((9 \times 5)=$ Units digit of $(45)=\mathbf{5}\)$.
- If the units digit of $\(7^X\)$ is 3 : Units digit of $\((3 \times 5)=$ Units digit of $(15)=\mathbf{5}\)$.
- If the units digit of $\(7^X\)$ is 1 : Units digit of $\((1 \times 5)=$ Units digit of $(5)=\mathbf{5}\)$.

In every possible case, the units digit of the entire expression is 5.
Therefore, the only possible units digit is 5 .