algebraic proof

Alright, here we go:

To prove that an expression is never positive, we just need to prove that the maximun value of the expression is smaller than zero.

We can just expand the expression. 

$3(x+1)(x+7)=3x^2+24x+21$

$(2x+5)^2=4x^2+20x+25$

$3x^2+24x+21- (4x^2+20x+25) = -x^2+4x-4$

Factoring out the negative one, we have:

$-(x^2-4x+4)$

Since

$(x^2-4x+4)$

is a perfect square, we can rewrite the expression like this:

$-(x-2)^2$

Since a square is always positive, and a negative of a square is negative, we proved that the original expression is negative, and always will be negative. 

I hope this answers your question and you have a wonderful day!