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\begin{array}{c}\phantom{\times99999}10990\\\underline{\times\phantom{99999}91003}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times99999}10990\\\underline{\times\phantom{99999}91003}\\\phantom{\times99999}32970\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 10990 with 3. Write the result 32970 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times99999}10990\\\underline{\times\phantom{99999}91003}\\\phantom{\times99999}32970\\\phantom{\times999999999}0\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 10990 with 0. Write the result 0 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times99999}10990\\\underline{\times\phantom{99999}91003}\\\phantom{\times99999}32970\\\phantom{\times999999999}0\phantom{9}\\\phantom{\times99999999}0\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 10990 with 0. Write the result 0 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times99999}10990\\\underline{\times\phantom{99999}91003}\\\phantom{\times99999}32970\\\phantom{\times999999999}0\phantom{9}\\\phantom{\times99999999}0\phantom{99}\\\phantom{\times99}10990\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 10990 with 1. Write the result 10990 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times99999}10990\\\underline{\times\phantom{99999}91003}\\\phantom{\times99999}32970\\\phantom{\times999999999}0\phantom{9}\\\phantom{\times99999999}0\phantom{99}\\\phantom{\times99}10990\phantom{999}\\\underline{\phantom{\times9}98910\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 10990 with 9. Write the result 98910 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times99999}10990\\\underline{\times\phantom{99999}91003}\\\phantom{\times99999}32970\\\phantom{\times999999999}0\phantom{9}\\\phantom{\times99999999}0\phantom{99}\\\phantom{\times99}10990\phantom{999}\\\underline{\phantom{\times9}98910\phantom{9999}}\\\phantom{\times}1000122970\end{array}
Now add the intermediate results to get final answer.