I'll multiply these two numbers systematically.

## Setting Up the Problem

Notice the structure:
- 123456789123456789 = 123456789 × 10⁹ + 123456789 = 123456789 × (10⁹ + 1)
- 987654321987654321 = 987654321 × (10⁹ + 1)

So the product is:
**123456789 × 987654321 × (10⁹ + 1)²**

## Step 1: Multiply 123456789 × 987654321

Using the known result:
123456789 × 987654321 = **121932631112635269**

## Step 2: Expand (10⁹ + 1)²

(10⁹ + 1)² = 10¹⁸ + 2×10⁹ + 1

## Step 3: Multiply 121932631112635269 × (10¹⁸ + 2×10⁹ + 1)

Let N = 121932631112635269

**N × 10¹⁸** = 121932631112635269000000000000000000

**N × 2×10⁹** = 243865262225270538000000000

**N × 1** = 121932631112635269

Now add them:

```
121932631112635269000000000000000000
+          243865262225270538000000000
+                   121932631112635269
```

Aligning and adding:

```
121932631112635269 000000000 000000000
                   243865262 225270538 000000000
                             121932631 112635269
```

Adding the middle and right sections:
- Rightmost 9 digits: **112635269**
- Middle 9 digits: 225270538 + 121932631 = **347203169**
- Left 9 digits: 243865262 + (carry 0) = **243865262**  
  Wait — checking: 225270538 + 121932631 = 347203169 (no carry)
- Left portion: 121932631112635269 + 243865262 (in the 10⁹ position)

Combining carefully:

```
121932631112635269
+       243865262  (×10⁹)
= 121932631356500531  [in the upper part]
```
121932631112635269 + 243865262000000000 = **121932631356500531000000000**... 

Let me redo the full addition cleanly:

```
121932631112635269 000000000 000000000
+         243865262 225270538 000000000
+                   121932631 112635269
─────────────────────────────────────
121932631356500531 347203169 112635269
```

## Final Answer

**123456789123456789 × 987654321987654321 = 121932631356500531347203169112635269**
