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\begin{array}{c}\phantom{\times}1234567\\\underline{\times\phantom{}34567}\\\end{array}
First line up the numbers vertically and match the places from the right like this.
\begin{array}{c}\phantom{\times}1234567\\\underline{\times\phantom{}34567}\\\phantom{\times}8641969\\\end{array}
Now multiply the first number with the 1^{st} digit in 2^{nd} number to get intermediate results. That is Multiply 1234567 with 7. Write the result 8641969 at the end leaving 0 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234567\\\underline{\times\phantom{}34567}\\\phantom{\times}8641969\\\phantom{\times}7407402\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 1234567 with 6. Write the result 7407402 at the end leaving 1 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234567\\\underline{\times\phantom{}34567}\\\phantom{\times}8641969\\\phantom{\times}7407402\phantom{9}\\\phantom{\times}6172835\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 1234567 with 5. Write the result 6172835 at the end leaving 2 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234567\\\underline{\times\phantom{}34567}\\\phantom{\times}8641969\\\phantom{\times}7407402\phantom{9}\\\phantom{\times}6172835\phantom{99}\\\phantom{\times}4938268\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 1234567 with 4. Write the result 4938268 at the end leaving 3 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234567\\\underline{\times\phantom{}34567}\\\phantom{\times}8641969\\\phantom{\times}7407402\phantom{9}\\\phantom{\times}6172835\phantom{99}\\\phantom{\times}4938268\phantom{999}\\\underline{\phantom{\times}3703701\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 1234567 with 3. Write the result 3703701 at the end leaving 4 spaces to the right like this.
\begin{array}{c}\phantom{\times}1234567\\\underline{\times\phantom{}34567}\\\phantom{\times}8641969\\\phantom{\times}7407402\phantom{9}\\\phantom{\times}6172835\phantom{99}\\\phantom{\times}4938268\phantom{999}\\\underline{\phantom{\times}3703701\phantom{9999}}\\\phantom{\times}-274395471\end{array}
Now add the intermediate results to get final answer.