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\begin{array}{c}\phantom{\times999999}258\\\underline{\times\phantom{99}1258963}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times999999}258\\\underline{\times\phantom{99}1258963}\\\phantom{\times999999}774\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 258 with 3. Write the result 774 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times999999}258\\\underline{\times\phantom{99}1258963}\\\phantom{\times999999}774\\\phantom{\times9999}1548\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 258 with 6. Write the result 1548 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times999999}258\\\underline{\times\phantom{99}1258963}\\\phantom{\times999999}774\\\phantom{\times9999}1548\phantom{9}\\\phantom{\times999}2322\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 258 with 9. Write the result 2322 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times999999}258\\\underline{\times\phantom{99}1258963}\\\phantom{\times999999}774\\\phantom{\times9999}1548\phantom{9}\\\phantom{\times999}2322\phantom{99}\\\phantom{\times99}2064\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 258 with 8. Write the result 2064 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times999999}258\\\underline{\times\phantom{99}1258963}\\\phantom{\times999999}774\\\phantom{\times9999}1548\phantom{9}\\\phantom{\times999}2322\phantom{99}\\\phantom{\times99}2064\phantom{999}\\\phantom{\times9}1290\phantom{9999}\\\end{array}
Now multiply the first number with the 5^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 258 with 5. Write the result 1290 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times999999}258\\\underline{\times\phantom{99}1258963}\\\phantom{\times999999}774\\\phantom{\times9999}1548\phantom{9}\\\phantom{\times999}2322\phantom{99}\\\phantom{\times99}2064\phantom{999}\\\phantom{\times9}1290\phantom{9999}\\\phantom{\times9}516\phantom{99999}\\\end{array}
Now multiply the first number with the 6^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 258 with 2. Write the result 516 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times999999}258\\\underline{\times\phantom{99}1258963}\\\phantom{\times999999}774\\\phantom{\times9999}1548\phantom{9}\\\phantom{\times999}2322\phantom{99}\\\phantom{\times99}2064\phantom{999}\\\phantom{\times9}1290\phantom{9999}\\\phantom{\times9}516\phantom{99999}\\\underline{\phantom{\times}258\phantom{999999}}\\\end{array}
Now multiply the first number with the 7^{th} digit in 2^{nd} number to get intermediate results. That is Multiply 258 with 1. Write the result 258 at the end leaving 6 spaces to the right like this.
\begin{array}{c}\phantom{\times999999}258\\\underline{\times\phantom{99}1258963}\\\phantom{\times999999}774\\\phantom{\times9999}1548\phantom{9}\\\phantom{\times999}2322\phantom{99}\\\phantom{\times99}2064\phantom{999}\\\phantom{\times9}1290\phantom{9999}\\\phantom{\times9}516\phantom{99999}\\\underline{\phantom{\times}258\phantom{999999}}\\\phantom{\times}324812454\end{array}
Now add the intermediate results to get final answer.