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\begin{array}{c}\phantom{\times99}377\\\underline{\times\phantom{99}145}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times99}377\\\underline{\times\phantom{99}145}\\\phantom{\times9}1885\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 377 with 5. Write the result 1885 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times99}377\\\underline{\times\phantom{99}145}\\\phantom{\times9}1885\\\phantom{\times}1508\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 377 with 4. Write the result 1508 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times99}377\\\underline{\times\phantom{99}145}\\\phantom{\times9}1885\\\phantom{\times}1508\phantom{9}\\\underline{\phantom{\times}377\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 377 with 1. Write the result 377 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times99}377\\\underline{\times\phantom{99}145}\\\phantom{\times9}1885\\\phantom{\times}1508\phantom{9}\\\underline{\phantom{\times}377\phantom{99}}\\\phantom{\times}54665\end{array}
Now add the intermediate results to get final answer.