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\begin{array}{c}\phantom{\times999}4567910\\\underline{\times\phantom{9999999}379}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times999}4567910\\\underline{\times\phantom{9999999}379}\\\phantom{\times99}41111190\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 4567910 with 9. Write the result 41111190 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times999}4567910\\\underline{\times\phantom{9999999}379}\\\phantom{\times99}41111190\\\phantom{\times9}31975370\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 4567910 with 7. Write the result 31975370 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times999}4567910\\\underline{\times\phantom{9999999}379}\\\phantom{\times99}41111190\\\phantom{\times9}31975370\phantom{9}\\\underline{\phantom{\times}13703730\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 4567910 with 3. Write the result 13703730 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times999}4567910\\\underline{\times\phantom{9999999}379}\\\phantom{\times99}41111190\\\phantom{\times9}31975370\phantom{9}\\\underline{\phantom{\times}13703730\phantom{99}}\\\phantom{\times}1731237890\end{array}
Now add the intermediate results to get final answer.