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\begin{array}{c}\phantom{\times9999}4578\\\underline{\times\phantom{9999}3214}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times9999}4578\\\underline{\times\phantom{9999}3214}\\\phantom{\times999}18312\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 4578 with 4. Write the result 18312 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}4578\\\underline{\times\phantom{9999}3214}\\\phantom{\times999}18312\\\phantom{\times999}4578\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 4578 with 1. Write the result 4578 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}4578\\\underline{\times\phantom{9999}3214}\\\phantom{\times999}18312\\\phantom{\times999}4578\phantom{9}\\\phantom{\times99}9156\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 4578 with 2. Write the result 9156 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}4578\\\underline{\times\phantom{9999}3214}\\\phantom{\times999}18312\\\phantom{\times999}4578\phantom{9}\\\phantom{\times99}9156\phantom{99}\\\underline{\phantom{\times}13734\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 4578 with 3. Write the result 13734 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}4578\\\underline{\times\phantom{9999}3214}\\\phantom{\times999}18312\\\phantom{\times999}4578\phantom{9}\\\phantom{\times99}9156\phantom{99}\\\underline{\phantom{\times}13734\phantom{999}}\\\phantom{\times}14713692\end{array}
Now add the intermediate results to get final answer.