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\begin{array}{c}\phantom{\times999}344\\\underline{\times\phantom{999}599}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times999}344\\\underline{\times\phantom{999}599}\\\phantom{\times99}3096\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 344 with 9. Write the result 3096 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times999}344\\\underline{\times\phantom{999}599}\\\phantom{\times99}3096\\\phantom{\times9}3096\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 344 with 9. Write the result 3096 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times999}344\\\underline{\times\phantom{999}599}\\\phantom{\times99}3096\\\phantom{\times9}3096\phantom{9}\\\underline{\phantom{\times}1720\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 344 with 5. Write the result 1720 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times999}344\\\underline{\times\phantom{999}599}\\\phantom{\times99}3096\\\phantom{\times9}3096\phantom{9}\\\underline{\phantom{\times}1720\phantom{99}}\\\phantom{\times}206056\end{array}
Now add the intermediate results to get final answer.