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\begin{array}{c}\phantom{\times999}5522526\\\underline{\times\phantom{999999}5823}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times999}5522526\\\underline{\times\phantom{999999}5823}\\\phantom{\times99}16567578\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 5522526 with 3. Write the result 16567578 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times999}5522526\\\underline{\times\phantom{999999}5823}\\\phantom{\times99}16567578\\\phantom{\times9}11045052\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 5522526 with 2. Write the result 11045052 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times999}5522526\\\underline{\times\phantom{999999}5823}\\\phantom{\times99}16567578\\\phantom{\times9}11045052\phantom{9}\\\phantom{\times}44180208\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 5522526 with 8. Write the result 44180208 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times999}5522526\\\underline{\times\phantom{999999}5823}\\\phantom{\times99}16567578\\\phantom{\times9}11045052\phantom{9}\\\phantom{\times}44180208\phantom{99}\\\underline{\phantom{\times}27612630\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 5522526 with 5. Write the result 27612630 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times999}5522526\\\underline{\times\phantom{999999}5823}\\\phantom{\times99}16567578\\\phantom{\times9}11045052\phantom{9}\\\phantom{\times}44180208\phantom{99}\\\underline{\phantom{\times}27612630\phantom{999}}\\\phantom{\times}2092897826\end{array}
Now add the intermediate results to get final answer.