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\begin{array}{c}\phantom{\times}321784\\\underline{\times\phantom{}345789}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times}321784\\\underline{\times\phantom{}345789}\\\phantom{\times}2896056\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 321784 with 9. Write the result 2896056 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times}321784\\\underline{\times\phantom{}345789}\\\phantom{\times}2896056\\\phantom{\times}2574272\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 321784 with 8. Write the result 2574272 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times}321784\\\underline{\times\phantom{}345789}\\\phantom{\times}2896056\\\phantom{\times}2574272\phantom{9}\\\phantom{\times}2252488\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 321784 with 7. Write the result 2252488 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times}321784\\\underline{\times\phantom{}345789}\\\phantom{\times}2896056\\\phantom{\times}2574272\phantom{9}\\\phantom{\times}2252488\phantom{99}\\\phantom{\times}1608920\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 321784 with 5. Write the result 1608920 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times}321784\\\underline{\times\phantom{}345789}\\\phantom{\times}2896056\\\phantom{\times}2574272\phantom{9}\\\phantom{\times}2252488\phantom{99}\\\phantom{\times}1608920\phantom{999}\\\phantom{\times}1287136\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 321784 with 4. Write the result 1287136 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times}321784\\\underline{\times\phantom{}345789}\\\phantom{\times}2896056\\\phantom{\times}2574272\phantom{9}\\\phantom{\times}2252488\phantom{99}\\\phantom{\times}1608920\phantom{999}\\\phantom{\times}1287136\phantom{9999}\\\underline{\phantom{\times}965352\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 321784 with 3. Write the result 965352 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times}321784\\\underline{\times\phantom{}345789}\\\phantom{\times}2896056\\\phantom{\times}2574272\phantom{9}\\\phantom{\times}2252488\phantom{99}\\\phantom{\times}1608920\phantom{999}\\\phantom{\times}1287136\phantom{9999}\\\underline{\phantom{\times}965352\phantom{99999}}\\\phantom{\times}-399782120\end{array}
Now add the intermediate results to get final answer.