Computing $999,999\cdot 222,222 + 333,333\cdot 333,334$ by hand.

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: $$\begin{align*} 3x(9x + 1) &= 333,333\cdot (999,999+1) \\ &= 333,333\cdot1,000,000 \\ &= \boxed{333,333,000,000} \end{align*}$$

It's about the same:

$$\begin{aligned}&999,999 \cdot 222,222 + 333,333 \cdot 333,334 \\ = &333,333 \cdot 666,666 + 333,333 \cdot 333,334 \\ = &333,333 \cdot 1,000,000\end{aligned}$$

An algebra free way: the expression is $$(10^6 -1)\bigg({2 \over 9}(10^6 - 1)\bigg) + {10^6 - 1 \over 3}{10^6 + 2 \over 3}$$ $$= {10^6 - 1 \over 3}\bigg({2 \over 3}(10^6 - 1) + {10^6 + 2 \over 3}\bigg)$$ $$= \bigg({10^6 - 1 \over 3}\bigg)10^6$$ $$= 333,333*1,000,000$$ $$= 333,333,000,000$$