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\begin{array}{c}\phantom{\times9999}88591\\\underline{\times\phantom{9999}55162}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times9999}88591\\\underline{\times\phantom{9999}55162}\\\phantom{\times999}177182\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 88591 with 2. Write the result 177182 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}88591\\\underline{\times\phantom{9999}55162}\\\phantom{\times999}177182\\\phantom{\times99}531546\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 88591 with 6. Write the result 531546 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}88591\\\underline{\times\phantom{9999}55162}\\\phantom{\times999}177182\\\phantom{\times99}531546\phantom{9}\\\phantom{\times99}88591\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 88591 with 1. Write the result 88591 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}88591\\\underline{\times\phantom{9999}55162}\\\phantom{\times999}177182\\\phantom{\times99}531546\phantom{9}\\\phantom{\times99}88591\phantom{99}\\\phantom{\times}442955\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 88591 with 5. Write the result 442955 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}88591\\\underline{\times\phantom{9999}55162}\\\phantom{\times999}177182\\\phantom{\times99}531546\phantom{9}\\\phantom{\times99}88591\phantom{99}\\\phantom{\times}442955\phantom{999}\\\underline{\phantom{\times}442955\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 88591 with 5. Write the result 442955 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times9999}88591\\\underline{\times\phantom{9999}55162}\\\phantom{\times999}177182\\\phantom{\times99}531546\phantom{9}\\\phantom{\times99}88591\phantom{99}\\\phantom{\times}442955\phantom{999}\\\underline{\phantom{\times}442955\phantom{9999}}\\\phantom{\times}591889446\end{array}
Now add the intermediate results to get final answer.