"entrenched into" is correct and can be used in written English
It is typically used with a metaphorical meaning to indicate something or someone is deeply entrenched in a particular belief, attitude, habit, or activity. For example, "He was entrenched into his old beliefs and refused to change."
It's a line that has been entrenched into every hardcore Twin Peaks fan's memory.
Some have raised concerns about the problem of child labor, which they say has deeply entrenched into the Indian system, exemplifying how socio-economic relations are negotiated within the Indian society.
Forms of segregation along lines of class is deeply entrenched into British society and has become even more pronounced in the last two decades, but no one talks about them.
Finally, PL spectra of our sample indicate that Nd3+ and Li+ ions are successfully entrenched into the CdS lattice producing enhanced luminescence properties.
The resistance against pull-out failure of the soil nails is provided by the part of soil nail that is entrenched into the ground behind the potential failure surface [6].
The aim of this study is to estimate the sediment residence time in the stream network of two small headwater catchments (Laval and Moulin) characterized by a badlands landscape entrenched into Jurassic black marls of the Southern French Prealps.
Depression has entrenched itself into who you are and how you think.
"The paid lobbyists under feminism have pretty much entrenched themselves into the system," one MRA told me.
A South Island doctor entrenched himself into kiwi pub folklore after he stitched up his own leg following a shark attack near Colac Bay.
In the last few decades, while we as a nation were busy creating and then cleaning up the messes of the world, the Brits quietly and quite unassumingly infiltrated our American airwaves and entrenched themselves into our hearts and minds, all in a masterful attempt to regain control of our precious land.
Without intervention, predatory, abusive behavior can become even more deeply entrenched in a young person as they grow into adulthood.
