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\begin{array}{c}\phantom{\times999}4599\\\underline{\times\phantom{999}1225}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times999}4599\\\underline{\times\phantom{999}1225}\\\phantom{\times99}22995\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 4599 with 5. Write the result 22995 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times999}4599\\\underline{\times\phantom{999}1225}\\\phantom{\times99}22995\\\phantom{\times99}9198\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 4599 with 2. Write the result 9198 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times999}4599\\\underline{\times\phantom{999}1225}\\\phantom{\times99}22995\\\phantom{\times99}9198\phantom{9}\\\phantom{\times9}9198\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 4599 with 2. Write the result 9198 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times999}4599\\\underline{\times\phantom{999}1225}\\\phantom{\times99}22995\\\phantom{\times99}9198\phantom{9}\\\phantom{\times9}9198\phantom{99}\\\underline{\phantom{\times}4599\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 4599 with 1. Write the result 4599 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times999}4599\\\underline{\times\phantom{999}1225}\\\phantom{\times99}22995\\\phantom{\times99}9198\phantom{9}\\\phantom{\times9}9198\phantom{99}\\\underline{\phantom{\times}4599\phantom{999}}\\\phantom{\times}5633775\end{array}
Now add the intermediate results to get final answer.