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\begin{array}{c}\phantom{\times}1234568\\\underline{\times\phantom{}1234568}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times}1234568\\\underline{\times\phantom{}1234568}\\\phantom{\times}9876544\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 1234568 with 8. Write the result 9876544 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234568\\\underline{\times\phantom{}1234568}\\\phantom{\times}9876544\\\phantom{\times}7407408\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 1234568 with 6. Write the result 7407408 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234568\\\underline{\times\phantom{}1234568}\\\phantom{\times}9876544\\\phantom{\times}7407408\phantom{9}\\\phantom{\times}6172840\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 1234568 with 5. Write the result 6172840 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234568\\\underline{\times\phantom{}1234568}\\\phantom{\times}9876544\\\phantom{\times}7407408\phantom{9}\\\phantom{\times}6172840\phantom{99}\\\phantom{\times}4938272\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 1234568 with 4. Write the result 4938272 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234568\\\underline{\times\phantom{}1234568}\\\phantom{\times}9876544\\\phantom{\times}7407408\phantom{9}\\\phantom{\times}6172840\phantom{99}\\\phantom{\times}4938272\phantom{999}\\\phantom{\times}3703704\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 1234568 with 3. Write the result 3703704 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234568\\\underline{\times\phantom{}1234568}\\\phantom{\times}9876544\\\phantom{\times}7407408\phantom{9}\\\phantom{\times}6172840\phantom{99}\\\phantom{\times}4938272\phantom{999}\\\phantom{\times}3703704\phantom{9999}\\\phantom{\times}2469136\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 1234568 with 2. Write the result 2469136 at the end leaving 5 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234568\\\underline{\times\phantom{}1234568}\\\phantom{\times}9876544\\\phantom{\times}7407408\phantom{9}\\\phantom{\times}6172840\phantom{99}\\\phantom{\times}4938272\phantom{999}\\\phantom{\times}3703704\phantom{9999}\\\phantom{\times}2469136\phantom{99999}\\\underline{\phantom{\times}1234568\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 1234568 with 1. Write the result 1234568 at the end leaving 6 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234568\\\underline{\times\phantom{}1234568}\\\phantom{\times}9876544\\\phantom{\times}7407408\phantom{9}\\\phantom{\times}6172840\phantom{99}\\\phantom{\times}4938272\phantom{999}\\\phantom{\times}3703704\phantom{9999}\\\phantom{\times}2469136\phantom{99999}\\\underline{\phantom{\times}1234568\phantom{999999}}\\\phantom{\times}-555243456\end{array}
Now add the intermediate results to get final answer.