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\begin{array}{c}\phantom{\times}4125963\\\underline{\times\phantom{}5814355}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times}4125963\\\underline{\times\phantom{}5814355}\\\phantom{\times}20629815\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 4125963 with 5. Write the result 20629815 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times}4125963\\\underline{\times\phantom{}5814355}\\\phantom{\times}20629815\\\phantom{\times}20629815\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 4125963 with 5. Write the result 20629815 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times}4125963\\\underline{\times\phantom{}5814355}\\\phantom{\times}20629815\\\phantom{\times}20629815\phantom{9}\\\phantom{\times}12377889\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 4125963 with 3. Write the result 12377889 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times}4125963\\\underline{\times\phantom{}5814355}\\\phantom{\times}20629815\\\phantom{\times}20629815\phantom{9}\\\phantom{\times}12377889\phantom{99}\\\phantom{\times}16503852\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 4125963 with 4. Write the result 16503852 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times}4125963\\\underline{\times\phantom{}5814355}\\\phantom{\times}20629815\\\phantom{\times}20629815\phantom{9}\\\phantom{\times}12377889\phantom{99}\\\phantom{\times}16503852\phantom{999}\\\phantom{\times}4125963\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 4125963 with 1. Write the result 4125963 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times}4125963\\\underline{\times\phantom{}5814355}\\\phantom{\times}20629815\\\phantom{\times}20629815\phantom{9}\\\phantom{\times}12377889\phantom{99}\\\phantom{\times}16503852\phantom{999}\\\phantom{\times}4125963\phantom{9999}\\\phantom{\times}33007704\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 4125963 with 8. Write the result 33007704 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times}4125963\\\underline{\times\phantom{}5814355}\\\phantom{\times}20629815\\\phantom{\times}20629815\phantom{9}\\\phantom{\times}12377889\phantom{99}\\\phantom{\times}16503852\phantom{999}\\\phantom{\times}4125963\phantom{9999}\\\phantom{\times}33007704\phantom{99999}\\\underline{\phantom{\times}20629815\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 4125963 with 5. Write the result 20629815 at the end leaving 6 spaces to the right like this.
\begin{array}{c}\phantom{\times}4125963\\\underline{\times\phantom{}5814355}\\\phantom{\times}20629815\\\phantom{\times}20629815\phantom{9}\\\phantom{\times}12377889\phantom{99}\\\phantom{\times}16503852\phantom{999}\\\phantom{\times}4125963\phantom{9999}\\\phantom{\times}33007704\phantom{99999}\\\underline{\phantom{\times}20629815\phantom{999999}}\\\phantom{\times}-1873716591\end{array}
Now add the intermediate results to get final answer.