Algebra 2(Math question)

Question

The trinomial 2x²+ 9x +9 has a coefficient greater than 1. Why do you multiply The coefficient 2 with the last term 9 to get 18 and then let the sum be 9 which is the middle term? What is the (Proof)? Why does it work that way? I need the (Proof) to explain why multiplying the first term and the last term is necessary and why does it also have be the sum of the middle term. I need (proof)

Answer 1

2x^2+9x+9

=(2x+3)(x+3)

use FOIL

2x^2+6x+3x+9

graphically it's an upward opening parabola with x intercepts -3 and -3/2

and y intercept = 9

vertex has x coordinate - 2.25, axis of symmetry is x= -2.25

complete the square

2(x^2+9x/2+(9/4)^2 +9 -2(9/4)^2

= 2(x+9/4)^2 -9/8

vertex = minimum point = (-9/4, -9/8)

unclear why you'd multiply 2 times 9

other than reversing operations when you factor out 2, before completing the square

maybe you meant divide 9 by 2?