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...

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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.**