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\begin{array}{c}\phantom{\times}555555555\\\underline{\times\phantom{9999}55555}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times}555555555\\\underline{\times\phantom{9999}55555}\\\phantom{\times999999999}-1517189521\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 555555555 with 5. Write the result -1517189521 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times}555555555\\\underline{\times\phantom{9999}55555}\\\phantom{\times999999999}-1517189521\\\phantom{\times99999999}-1517189521\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 555555555 with 5. Write the result -1517189521 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times}555555555\\\underline{\times\phantom{9999}55555}\\\phantom{\times999999999}-1517189521\\\phantom{\times99999999}-1517189521\phantom{9}\\\phantom{\times9999999}-1517189521\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 555555555 with 5. Write the result -1517189521 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times}555555555\\\underline{\times\phantom{9999}55555}\\\phantom{\times999999999}-1517189521\\\phantom{\times99999999}-1517189521\phantom{9}\\\phantom{\times9999999}-1517189521\phantom{99}\\\phantom{\times999999}-1517189521\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 555555555 with 5. Write the result -1517189521 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times}555555555\\\underline{\times\phantom{9999}55555}\\\phantom{\times999999999}-1517189521\\\phantom{\times99999999}-1517189521\phantom{9}\\\phantom{\times9999999}-1517189521\phantom{99}\\\phantom{\times999999}-1517189521\phantom{999}\\\underline{\phantom{\times99999}-1517189521\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 555555555 with 5. Write the result -1517189521 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times}555555555\\\underline{\times\phantom{9999}55555}\\\phantom{\times999999999}-1517189521\\\phantom{\times99999999}-1517189521\phantom{9}\\\phantom{\times9999999}-1517189521\phantom{99}\\\phantom{\times999999}-1517189521\phantom{999}\\\underline{\phantom{\times99999}-1517189521\phantom{9999}}\\\phantom{\times}253868969\end{array}
Now add the intermediate results to get final answer.