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The expression x^2 + 2x + 5 can be expressed in the form (x+p)^2 + q as follows:
x^2 + 2x + 5
=> x^2 + 2x + 1 + 4
=> (x + 1)^2 + 4
The term (x + 1)^2 is a square and always positive. 4 is also a positive integer. The sum of two positive terms is always positive.
This proves that x^2 + 2x + 5 is always positive.
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