The expression x^2 + 2bx + c has a positive v | vedclass

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(D) The given expression is $f(x) = x^2 + 2bx + c$.By completing the square,we can write:$f(x) = (x^2 + 2bx + b^2) - b^2 + c$$f(x) = (x + b)^2 + (c -

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$b^2 < c$

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(D) The given expression is $f(x) = x^2 + 2bx + c$.By completing the square,we can write:$f(x) = (x^2 + 2bx + b^2) - b^2 + c$$f(x) = (x + b)^2 + (c - b^2)$Since $(x + b)^2 \ge 0$ for all real $x$,the expression $f(x)$ will always be positive if the constant term $(c - b^2)$ is strictly greater than $0$.Therefore,$c - b^2 > 0$,which implies $c > b^2$ or $b^2 < c$.

The expression $x^2 + 2bx + c$ has a positive value for all real $x$ if:

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