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\begin{array}{c}\phantom{\times999}42\\\underline{\times\phantom{9}2369}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times999}42\\\underline{\times\phantom{9}2369}\\\phantom{\times99}378\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 42 with 9. Write the result 378 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times999}42\\\underline{\times\phantom{9}2369}\\\phantom{\times99}378\\\phantom{\times9}252\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 42 with 6. Write the result 252 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times999}42\\\underline{\times\phantom{9}2369}\\\phantom{\times99}378\\\phantom{\times9}252\phantom{9}\\\phantom{\times}126\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 42 with 3. Write the result 126 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times999}42\\\underline{\times\phantom{9}2369}\\\phantom{\times99}378\\\phantom{\times9}252\phantom{9}\\\phantom{\times}126\phantom{99}\\\underline{\phantom{\times}84\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 42 with 2. Write the result 84 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times999}42\\\underline{\times\phantom{9}2369}\\\phantom{\times99}378\\\phantom{\times9}252\phantom{9}\\\phantom{\times}126\phantom{99}\\\underline{\phantom{\times}84\phantom{999}}\\\phantom{\times}99498\end{array}
Now add the intermediate results to get final answer.