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\begin{array}{c}\phantom{\times}999999999\\\underline{\times\phantom{}876543210}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times}999999999\\\underline{\times\phantom{}876543210}\\\phantom{\times}0\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 999999999 with 0. Write the result 0 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times}999999999\\\underline{\times\phantom{}876543210}\\\phantom{\times}0\\\phantom{\times}999999999\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 999999999 with 1. Write the result 999999999 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times}999999999\\\underline{\times\phantom{}876543210}\\\phantom{\times}0\\\phantom{\times}999999999\phantom{9}\\\phantom{\times}1999999998\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 999999999 with 2. Write the result 1999999998 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times}999999999\\\underline{\times\phantom{}876543210}\\\phantom{\times}0\\\phantom{\times}999999999\phantom{9}\\\phantom{\times}1999999998\phantom{99}\\\phantom{\times}-1294967299\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 999999999 with 3. Write the result -1294967299 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times}999999999\\\underline{\times\phantom{}876543210}\\\phantom{\times}0\\\phantom{\times}999999999\phantom{9}\\\phantom{\times}1999999998\phantom{99}\\\phantom{\times}-1294967299\phantom{999}\\\phantom{\times}-294967300\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 999999999 with 4. Write the result -294967300 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times}999999999\\\underline{\times\phantom{}876543210}\\\phantom{\times}0\\\phantom{\times}999999999\phantom{9}\\\phantom{\times}1999999998\phantom{99}\\\phantom{\times}-1294967299\phantom{999}\\\phantom{\times}-294967300\phantom{9999}\\\phantom{\times}705032699\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 999999999 with 5. Write the result 705032699 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times}999999999\\\underline{\times\phantom{}876543210}\\\phantom{\times}0\\\phantom{\times}999999999\phantom{9}\\\phantom{\times}1999999998\phantom{99}\\\phantom{\times}-1294967299\phantom{999}\\\phantom{\times}-294967300\phantom{9999}\\\phantom{\times}705032699\phantom{99999}\\\phantom{\times}1705032698\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 999999999 with 6. Write the result 1705032698 at the end leaving 6 spaces to the right like this.
\begin{array}{c}\phantom{\times}999999999\\\underline{\times\phantom{}876543210}\\\phantom{\times}0\\\phantom{\times}999999999\phantom{9}\\\phantom{\times}1999999998\phantom{99}\\\phantom{\times}-1294967299\phantom{999}\\\phantom{\times}-294967300\phantom{9999}\\\phantom{\times}705032699\phantom{99999}\\\phantom{\times}1705032698\phantom{999999}\\\phantom{\times}-1589934599\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 999999999 with 7. Write the result -1589934599 at the end leaving 7 spaces to the right like this.
\begin{array}{c}\phantom{\times}999999999\\\underline{\times\phantom{}876543210}\\\phantom{\times}0\\\phantom{\times}999999999\phantom{9}\\\phantom{\times}1999999998\phantom{99}\\\phantom{\times}-1294967299\phantom{999}\\\phantom{\times}-294967300\phantom{9999}\\\phantom{\times}705032699\phantom{99999}\\\phantom{\times}1705032698\phantom{999999}\\\phantom{\times}-1589934599\phantom{9999999}\\\underline{\phantom{\times}-589934600\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 999999999 with 8. Write the result -589934600 at the end leaving 8 spaces to the right like this.
\begin{array}{c}\phantom{\times}999999999\\\underline{\times\phantom{}876543210}\\\phantom{\times}0\\\phantom{\times}999999999\phantom{9}\\\phantom{\times}1999999998\phantom{99}\\\phantom{\times}-1294967299\phantom{999}\\\phantom{\times}-294967300\phantom{9999}\\\phantom{\times}705032699\phantom{99999}\\\phantom{\times}1705032698\phantom{999999}\\\phantom{\times}-1589934599\phantom{9999999}\\\underline{\phantom{\times}-589934600\phantom{99999999}}\\\phantom{\times}-1844074730\end{array}
Now add the intermediate results to get final answer.