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\begin{array}{c}\phantom{\times}12345\\\underline{\times\phantom{}678910}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times}12345\\\underline{\times\phantom{}678910}\\\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 12345 with 0. Write the result 0 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times}12345\\\underline{\times\phantom{}678910}\\\phantom{\times}0\\\phantom{\times}12345\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 12345 with 1. Write the result 12345 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times}12345\\\underline{\times\phantom{}678910}\\\phantom{\times}0\\\phantom{\times}12345\phantom{9}\\\phantom{\times}111105\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 12345 with 9. Write the result 111105 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times}12345\\\underline{\times\phantom{}678910}\\\phantom{\times}0\\\phantom{\times}12345\phantom{9}\\\phantom{\times}111105\phantom{99}\\\phantom{\times}98760\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 12345 with 8. Write the result 98760 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times}12345\\\underline{\times\phantom{}678910}\\\phantom{\times}0\\\phantom{\times}12345\phantom{9}\\\phantom{\times}111105\phantom{99}\\\phantom{\times}98760\phantom{999}\\\phantom{\times}86415\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 12345 with 7. Write the result 86415 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times}12345\\\underline{\times\phantom{}678910}\\\phantom{\times}0\\\phantom{\times}12345\phantom{9}\\\phantom{\times}111105\phantom{99}\\\phantom{\times}98760\phantom{999}\\\phantom{\times}86415\phantom{9999}\\\underline{\phantom{\times}74070\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 12345 with 6. Write the result 74070 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times}12345\\\underline{\times\phantom{}678910}\\\phantom{\times}0\\\phantom{\times}12345\phantom{9}\\\phantom{\times}111105\phantom{99}\\\phantom{\times}98760\phantom{999}\\\phantom{\times}86415\phantom{9999}\\\underline{\phantom{\times}74070\phantom{99999}}\\\phantom{\times}-208790642\end{array}
Now add the intermediate results to get final answer.