Computing $999,999\cdot 222,222 + 333,333\cdot 333,334$ by hand. I got this question from a last year's olympiad paper.

Compute $999,999\cdot 222,222 + 333,333\cdot 333,334$.

Is there an approach to this by using pen-and-paper?
EDIT Working through on paper made me figure out the answer. Posted below.
I'd now like to see other methods. Thank you.
 A: My immediate thought was to compute this as $$1,000,000\times222,222 - 222,222+111,111\times1,000,002$$
A: It's about the same:
$$\begin{aligned}&999,999 \cdot 222,222 + 333,333 \cdot 333,334 \\ = &333,333 \cdot 666,666 + 333,333 \cdot 333,334 \\ = &333,333 \cdot 1,000,000\end{aligned}$$
A: $$999,999\cdot 222,222 + 333,333\cdot 333,334=333,333\cdot 666,666 + 333,333\cdot 333,334$$
$$=333,333 \cdot 1,000,000$$
A: An algebra free way: the expression is
$$(10^6 -1)\bigg({2 \over 9}(10^6 - 1)\bigg) + {10^6 - 1 \over 3}{10^6 + 2 \over 3}$$
$$= {10^6 - 1 \over 3}\bigg({2 \over 3}(10^6 - 1) + {10^6 + 2 \over 3}\bigg)$$
$$= \bigg({10^6 - 1 \over 3}\bigg)10^6$$
$$= 333,333*1,000,000$$
$$= 333,333,000,000$$
A: My first (useful) though was factoring out 333,333:
$$\begin{align*}
&999,999*222,222+333,333*333,334 =\\
&= 333,333*(3*222,222+333,334) =\\
&= 333,333*(666,666+333,334) =\\
&= 333,333*1,000,000 =\\
&= 333,333,000,000
\end{align*}$$
A: My observation suggests that we may simplify it somewhat like this assuming $x = 111,111$:
$$\begin{align*}
9x\cdot2x+3x(3x + 1) &= 9x^2 +18x^2 +  3x
\\ &= 27x^2 + 3x
\\ &= 3x(9x + 1)
\end{align*}$$
So this means:
$$\begin{align*}
3x(9x + 1) &= 333,333\cdot (999,999+1)
\\ &= 333,333\cdot1,000,000
\\ &= \boxed{333,333,000,000}
\end{align*}$$
