Final answer:
To prove that 3(x+1)(x+7)-(2x+5)² is never positive, we can expand and simplify the expression to -x²+4x-4, which is always negative.
Step-by-step explanation:
To prove that 3(x+1)(x+7)-(2x+5)² is never positive, we can use algebraic manipulation and properties of quadratic equations. Let's expand and simplify the expression:
3(x+1)(x+7)-(2x+5)² = 3(x²+8x+7)-(4x²+20x+25) = 3x²+24x+21-4x²-20x-25 = -x²+4x-4
Since the leading coefficient of the quadratic term is negative (-1), the graph of this equation will be a downward-opening parabola. Therefore, the quadratic expression is never positive.