The product is 121,932,631,112,635,269 (about 1.2193 × 10^17).
You can check it with a shortcut. Write 123,456,789 as 10 × 12,345,679 − 1, and note that 987,654,321 = 10^9 − 12,345,679. Squaring 123,456,789 gives 15,241,578,750,190,521, and from that you can get 12,345,679² = 152,415,789,971,041. Those pieces combine to the same 17-digit result.
This is exact integer arithmetic, so there's no rounding. Your own calculator or a spreadsheet will confirm it too, though most calculators will show it in scientific notation unless you use a program that handles big integers.
If anyone wants to try a different big multiplication by hand, the same split-into-near-powers-of-ten trick works well for numbers close to round values.
