To multiply $123,456,789,123,456,789$ by $987,654,321,987,654,321$ exactly, we can use the property of repeating patterns or algebraic expansion.

### 1. Identify the Pattern
Note that:
$A = 123,456,789,123,456,789 = (123,456,789 \times 10^9) + 123,456,789$
$B = 987,654,321,987,654,321 = (987,654,321 \times 10^9) + 987,654,321$

Let $x = 123,456,789$ and $y = 987,654,321$.
The expression is $(x \cdot 10^9 + x)(y \cdot 10^9 + y)$.
Factoring this gives: $x(10^9 + 1) \cdot y(10^9 + 1) = (xy)(10^9 + 1)^2$
$= (xy)(10^{18} + 2 \cdot 10^9 + 1)$

### 2. Calculate the base product ($xy$)
First, multiply $123,456,789 \times 987,654,321$:
$123,456,789 \times 987,654,321 = 121,932,631,112,635,269$

### 3. Expand the expression
Now multiply $121,932,631,112,635,269$ by $(10^{18} + 2 \cdot 10^9 + 1)$:

1.  **Term 1 ($xy \cdot 10^{18}$):**
    $121,932,631,112,635,269,000,000,000,000,000,000$
2.  **Term 2 ($xy \cdot 2 \cdot 10^9$):**
    $243,865,262,225,270,538,000,000,000$ (shifted 9 places)
3.  **Term 3 ($xy \cdot 1$):**
    $121,932,631,112,635,269$

### 4. Sum the terms
Adding them carefully by alignment:

```text
  121932631112635269000000000000000000 (Term 1)
+        243865262225270538000000000   (Term 2)
+                   121932631112635269 (Term 3)
--------------------------------------
  121932631356500531337837899112635269
```

**Final Answer:**
The exact product is:
**121,932,631,356,500,531,337,837,899,112,635,269**
