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The phrase "deep ingrained" is correct and can be used in written English.
It is used to describe something that is firmly established or deeply rooted in someone's beliefs, behavior, or personality. Example: The fear of failure was deep ingrained in her, causing her to avoid taking risks or trying new things.
Exact(4)
I understood why Derbyshire had been nagging at my subconscious: this bleak, craggy up-and-down terrain and damp climate was a deep, ingrained part of me.
True, the deep, ingrained rivalry and mistrust between Libya's military-security apparatuses is likely to give rise to splits that will be a key factor in undermining the regime.
He added: "The key and essential question here is: how do you tackle these deep ingrained problems of poverty?
The value of such study is being realized increasingly but do we not all know the business man who can be considered only as a mere slap-dash empiricist, with a breezy confidence in hunches and a deep, ingrained dislike of statistical tables and bar charts?
Similar(56)
Since Mohammed Abu Khdeir was killed, the hatred has become more personalized, wider spread, and deeper ingrained.
Yet, far from this, Merkel's refusal to provide strong leadership to demand the much closer fiscal integration that monetary union requires, and effective, German-led control, shows how deep-ingrained is the German political classes' instinct against leadership beyond their national borders.
The Thais, a bit like the Balinese, are coexisting with other cultures, but they've managed to maintain and preserve their values in a very deep and ingrained way.
Schipper follows with a broad examination of the ways in which religion can be included in DRR, and Crabtree finishes the section with a case of the deep, culturally ingrained responses of people surrounding the River Kosi when it flooded in 2008.
After differentiating between the superficial anachronism found in the casualty figures on the army banners and the deeper anachronism ingrained in the painting's contemporary context, he posits that the latter type is less a superimposition of one historical event over another and more an acknowledgement of the recursive nature of history.
The song is ingrained deep in me.
"It's ingrained deep in my bones not to ask taxpayers to pay for this unless we absolutely had to".
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Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.

Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com