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\begin{array}{c}\phantom{\times999}107129\\\underline{\times\phantom{99999}1757}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times999}107129\\\underline{\times\phantom{99999}1757}\\\phantom{\times999}749903\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 107129 with 7. Write the result 749903 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times999}107129\\\underline{\times\phantom{99999}1757}\\\phantom{\times999}749903\\\phantom{\times99}535645\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 107129 with 5. Write the result 535645 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times999}107129\\\underline{\times\phantom{99999}1757}\\\phantom{\times999}749903\\\phantom{\times99}535645\phantom{9}\\\phantom{\times9}749903\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 107129 with 7. Write the result 749903 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times999}107129\\\underline{\times\phantom{99999}1757}\\\phantom{\times999}749903\\\phantom{\times99}535645\phantom{9}\\\phantom{\times9}749903\phantom{99}\\\underline{\phantom{\times}107129\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 107129 with 1. Write the result 107129 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times999}107129\\\underline{\times\phantom{99999}1757}\\\phantom{\times999}749903\\\phantom{\times99}535645\phantom{9}\\\phantom{\times9}749903\phantom{99}\\\underline{\phantom{\times}107129\phantom{999}}\\\phantom{\times}188225653\end{array}
Now add the intermediate results to get final answer.