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\begin{array}{c}\phantom{\times9999999}432\\\underline{\times\phantom{999}3623499}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times9999999}432\\\underline{\times\phantom{999}3623499}\\\phantom{\times999999}3888\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 432 with 9. Write the result 3888 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times9999999}432\\\underline{\times\phantom{999}3623499}\\\phantom{\times999999}3888\\\phantom{\times99999}3888\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 432 with 9. Write the result 3888 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times9999999}432\\\underline{\times\phantom{999}3623499}\\\phantom{\times999999}3888\\\phantom{\times99999}3888\phantom{9}\\\phantom{\times9999}1728\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 432 with 4. Write the result 1728 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times9999999}432\\\underline{\times\phantom{999}3623499}\\\phantom{\times999999}3888\\\phantom{\times99999}3888\phantom{9}\\\phantom{\times9999}1728\phantom{99}\\\phantom{\times999}1296\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 432 with 3. Write the result 1296 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times9999999}432\\\underline{\times\phantom{999}3623499}\\\phantom{\times999999}3888\\\phantom{\times99999}3888\phantom{9}\\\phantom{\times9999}1728\phantom{99}\\\phantom{\times999}1296\phantom{999}\\\phantom{\times999}864\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 432 with 2. Write the result 864 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times9999999}432\\\underline{\times\phantom{999}3623499}\\\phantom{\times999999}3888\\\phantom{\times99999}3888\phantom{9}\\\phantom{\times9999}1728\phantom{99}\\\phantom{\times999}1296\phantom{999}\\\phantom{\times999}864\phantom{9999}\\\phantom{\times9}2592\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 432 with 6. Write the result 2592 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times9999999}432\\\underline{\times\phantom{999}3623499}\\\phantom{\times999999}3888\\\phantom{\times99999}3888\phantom{9}\\\phantom{\times9999}1728\phantom{99}\\\phantom{\times999}1296\phantom{999}\\\phantom{\times999}864\phantom{9999}\\\phantom{\times9}2592\phantom{99999}\\\underline{\phantom{\times}1296\phantom{999999}}\\\end{array}
Now multiply the first number with the 7^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 432 with 3. Write the result 1296 at the end leaving 6 spaces to the right like this.
\begin{array}{c}\phantom{\times9999999}432\\\underline{\times\phantom{999}3623499}\\\phantom{\times999999}3888\\\phantom{\times99999}3888\phantom{9}\\\phantom{\times9999}1728\phantom{99}\\\phantom{\times999}1296\phantom{999}\\\phantom{\times999}864\phantom{9999}\\\phantom{\times9}2592\phantom{99999}\\\underline{\phantom{\times}1296\phantom{999999}}\\\phantom{\times}1565351568\end{array}
Now add the intermediate results to get final answer.