We compute both sides, the estimate we have to show is then
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![{\displaystyle {}2x^{2}+2x+1\geq x^{2}+4x+4\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9d72ec9329c3203523994e65a1e5409e24be5b17)
For real numbers we can add on both sides, so the estimate is equivalent with
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![{\displaystyle {}x^{2}-2x-3\geq 0\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d8d1096be9cdbc72e49e77832346594d0ab10fb8)
We write the left hand side as
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![{\displaystyle {}x^{2}-2x-3=x^{2}-2x+1-1-3=(x-1)^{2}-4\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d8f38ca8652974b08cb68dd11db71949bd8d37f)
For
we have
and therefore
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![{\displaystyle {}(x-1)^{2}\geq 2(x-1)\geq 2^{2}=4\,.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/22c5c3d39d870c0ceb82a9919c3b29a71bd6813f)
So for
we get the estimate
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![{\displaystyle {}(x-1)^{2}-4\geq 0\,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dadad2a414a22432e9f8131741ee82f4557617b2)
and hence the original estimate.