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\begin{array}{c}\phantom{\times}555297\\\underline{\times\phantom{}4679}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times}555297\\\underline{\times\phantom{}4679}\\\phantom{\times}4997673\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 555297 with 9. Write the result 4997673 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times}555297\\\underline{\times\phantom{}4679}\\\phantom{\times}4997673\\\phantom{\times}3887079\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 555297 with 7. Write the result 3887079 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times}555297\\\underline{\times\phantom{}4679}\\\phantom{\times}4997673\\\phantom{\times}3887079\phantom{9}\\\phantom{\times}3331782\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 555297 with 6. Write the result 3331782 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times}555297\\\underline{\times\phantom{}4679}\\\phantom{\times}4997673\\\phantom{\times}3887079\phantom{9}\\\phantom{\times}3331782\phantom{99}\\\underline{\phantom{\times}2221188\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 555297 with 4. Write the result 2221188 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times}555297\\\underline{\times\phantom{}4679}\\\phantom{\times}4997673\\\phantom{\times}3887079\phantom{9}\\\phantom{\times}3331782\phantom{99}\\\underline{\phantom{\times}2221188\phantom{999}}\\\phantom{\times}-1696732633\end{array}
Now add the intermediate results to get final answer.