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\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 9. Write the result 8999991 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\end{array}
Now multiply the first number with the 2^{nd} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 8. Write the result 7999992 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\end{array}
Now multiply the first number with the 3^{rd} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 7. Write the result 6999993 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\end{array}
Now multiply the first number with the 4^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 6. Write the result 5999994 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\end{array}
Now multiply the first number with the 5^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 5. Write the result 4999995 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\phantom{\times}3999996\phantom{99999}\\\end{array}
Now multiply the first number with the 6^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 4. Write the result 3999996 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\phantom{\times}3999996\phantom{99999}\\\phantom{\times}2999997\phantom{999999}\\\end{array}
Now multiply the first number with the 7^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 3. Write the result 2999997 at the end leaving 6 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\phantom{\times}3999996\phantom{99999}\\\phantom{\times}2999997\phantom{999999}\\\phantom{\times}1999998\phantom{9999999}\\\end{array}
Now multiply the first number with the 8^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 2. Write the result 1999998 at the end leaving 7 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\phantom{\times}3999996\phantom{99999}\\\phantom{\times}2999997\phantom{999999}\\\phantom{\times}1999998\phantom{9999999}\\\underline{\phantom{\times}999999\phantom{99999999}}\\\end{array}
Now multiply the first number with the 9^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 999999 with 1. Write the result 999999 at the end leaving 8 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}999999\\\underline{\times\phantom{9}123456789}\\\phantom{\times999}8999991\\\phantom{\times99}7999992\phantom{9}\\\phantom{\times9}6999993\phantom{99}\\\phantom{\times}5999994\phantom{999}\\\phantom{\times}4999995\phantom{9999}\\\phantom{\times}3999996\phantom{99999}\\\phantom{\times}2999997\phantom{999999}\\\phantom{\times}1999998\phantom{9999999}\\\underline{\phantom{\times}999999\phantom{99999999}}\\\phantom{\times}2125586987\end{array}
Now add the intermediate results to get final answer.