To solve the given problem, let's break down the sets $A$ and $B$ one by one and analyze their properties.

Set $A$:

[ A = {x \in \mathbb{R} \mid [x + 3] + [x + 5] \leq 3} ] Here, $[t]$ denotes the greatest integer less than or equal to $t$.

1. Understanding the expression $[x + 3] + [x + 5]$:

   • The greatest integer function $[x + 3]$ and $[x + 5]$ represent the greatest integers less than or equal to $x + 3$ and $x + 5$, respectively.

2. Range Analysis:

   • If $x$ is an integer, then $[x + 3]$ and $[x + 5]$ are simply $x + 3$ and $x + 5$.
   • Consider non-integer $x$ in intervals $n \leq x < n+1$ where $n$ is an integer.

3. Considering specific values of $x$:

   • For $x = 0$: $[3] + [5] = 3 + 5 = 8$ (not possible as $8 > 3$)
   • For $x = -3.5$: $[x + 3] = [0.5] = 0$, $[x + 5] = [1.5] = 1$, hence $0 + 1 = 1 \leq 3$.

4. Possible values for $x$:

   • We must have $[x + 3] + [x + 5] \leq 3$, so $x$ must be around intervals where $x+3$ and $x+5$ are near the integer values of 0, 1, 2, etc.

5. Conclude $A$:

   • The values of $x$ that satisfy this inequality are $-3.5 \leq x < -2$ or $-3 \leq x < -2.5$.

Set $B$:

$B = \{x \in \mathbb{R} \mid (3^x) \cdot |x + 3| < 1 \}$

1. Analyzing $(3^x) \cdot |x + 3| < 1$:

   • $3^x$ is an exponential function that grows rapidly.
   • The inequality implies $3^x < \frac{1}{|x + 3|}$.

2. Behavior based on $x$:

   • If $x$ is very negative, $3^x$ becomes small, and the inequality might be satisfied.
   • If $x$ is close to $-3$, then $|x + 3|$ is small, making $\frac{1}{|x + 3|}$ large, possibly satisfying the inequality.

3. Exact Solutions:

   • For $x = -3$, $|x + 3| = 0$, which does not satisfy the inequality (division by zero).
   • Consider small deviations around $x = -3$.

Intersection and Subset Analysis:

• $A \cap B$:

  • There might be a common interval where both inequalities are satisfied.
  • $A$ includes values like $x$ near $-3.5$, which can be verified for $B$.

• Subset Relationships:

  • If all elements of $B$ are within $A$, $B \subset A$.
  • Conversely, $A \subset B$ might be true if all elements of $A$ satisfy the inequality for $B$.

Conclusion:

After analyzing the behavior and checking possible values of $x$, the best fit is that $B \subset A$.

Thus, the correct answer is $\boxed{3}$ $B \subset A$.

Would you like more details or have any questions? Here are some related questions:

1. How does the greatest integer function behave for non-integer inputs?
2. What are the general properties of exponential functions like $3^x$?
3. How does the condition $|x+3|$ affect the inequality in set $B$?
4. What happens if $x$ approaches $-3$ from the left or right?
5. Can the behavior of $B$ change if $3^x$ were replaced with another base?

Tip: Always consider the specific intervals where the greatest integer function changes its value when solving inequalities.