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\begin{array}{c}\phantom{\times}936977\\\underline{\times\phantom{}453731}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times}936977\\\underline{\times\phantom{}453731}\\\phantom{\times}936977\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 936977 with 1. Write the result 936977 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times}936977\\\underline{\times\phantom{}453731}\\\phantom{\times}936977\\\phantom{\times}2810931\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 936977 with 3. Write the result 2810931 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times}936977\\\underline{\times\phantom{}453731}\\\phantom{\times}936977\\\phantom{\times}2810931\phantom{9}\\\phantom{\times}6558839\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 936977 with 7. Write the result 6558839 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times}936977\\\underline{\times\phantom{}453731}\\\phantom{\times}936977\\\phantom{\times}2810931\phantom{9}\\\phantom{\times}6558839\phantom{99}\\\phantom{\times}2810931\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 936977 with 3. Write the result 2810931 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times}936977\\\underline{\times\phantom{}453731}\\\phantom{\times}936977\\\phantom{\times}2810931\phantom{9}\\\phantom{\times}6558839\phantom{99}\\\phantom{\times}2810931\phantom{999}\\\phantom{\times}4684885\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 936977 with 5. Write the result 4684885 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times}936977\\\underline{\times\phantom{}453731}\\\phantom{\times}936977\\\phantom{\times}2810931\phantom{9}\\\phantom{\times}6558839\phantom{99}\\\phantom{\times}2810931\phantom{999}\\\phantom{\times}4684885\phantom{9999}\\\underline{\phantom{\times}3747908\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 936977 with 4. Write the result 3747908 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times}936977\\\underline{\times\phantom{}453731}\\\phantom{\times}936977\\\phantom{\times}2810931\phantom{9}\\\phantom{\times}6558839\phantom{99}\\\phantom{\times}2810931\phantom{999}\\\phantom{\times}4684885\phantom{9999}\\\underline{\phantom{\times}3747908\phantom{99999}}\\\phantom{\times}-66251117\end{array}
Now add the intermediate results to get final answer.