When integrating websites within other websites, consider using an iframe
element, as shown below:
<iframe src="http://www.bing.com"></iframe>
For more details, visit w3schools
Google
Please note that the above method will not work for Google due to their terms of service:
According to Google's terms, you are not allowed to frame, cache, or modify the search results generated by Google without their permission.
If you wish to integrate Google search functionality into your website, consider using the Google API. More information is available here Google Api
Bing
For embedding search results from Bing, check out the Bing API. It offers up to 5000 free queries per day.