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\begin{array}{c}\phantom{\times9999}12\\\underline{\times\phantom{9}12123}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times9999}12\\\underline{\times\phantom{9}12123}\\\phantom{\times9999}36\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 12 with 3. Write the result 36 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}12\\\underline{\times\phantom{9}12123}\\\phantom{\times9999}36\\\phantom{\times999}24\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 12 with 2. Write the result 24 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}12\\\underline{\times\phantom{9}12123}\\\phantom{\times9999}36\\\phantom{\times999}24\phantom{9}\\\phantom{\times99}12\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 12 with 1. Write the result 12 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}12\\\underline{\times\phantom{9}12123}\\\phantom{\times9999}36\\\phantom{\times999}24\phantom{9}\\\phantom{\times99}12\phantom{99}\\\phantom{\times9}24\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 12 with 2. Write the result 24 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}12\\\underline{\times\phantom{9}12123}\\\phantom{\times9999}36\\\phantom{\times999}24\phantom{9}\\\phantom{\times99}12\phantom{99}\\\phantom{\times9}24\phantom{999}\\\underline{\phantom{\times}12\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 12 with 1. Write the result 12 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}12\\\underline{\times\phantom{9}12123}\\\phantom{\times9999}36\\\phantom{\times999}24\phantom{9}\\\phantom{\times99}12\phantom{99}\\\phantom{\times9}24\phantom{999}\\\underline{\phantom{\times}12\phantom{9999}}\\\phantom{\times}145476\end{array}
Now add the intermediate results to get final answer.