Step 1: Foil (multiply) the right hand side of the equation
y= (x+5)(x+3)
rArr y= x^2 + 5x + 3x + 15
=> color(red)(y= x^2 + 8x + 15)
Step 2: We can write the vertex form by several methods
Reminder: vertex form is color(blue)(y= a(x-h)^2 + k)
=> Method 1: By completing square
=> color(red)(y= x^2 + 8x + 15) => re-write
We make a perfect trinomial in the form of
=> a^2 -2ab +b^2 = (a-b)^2
=> a^2 +2ab+b^2 = (a+b)^2
y = (x^2 + 8x+color(green)16) color(green)(-16)+15
16= [1/2 (8)]^2
y= (x+4)^2 -1 Vertex form completed
=> **Method 2: Using formula **
h= x_(vertex)= -b/(2a)
k= y_(vertex)= y(-b/(ab))
From this=> color(red)(y= x^2 + 8x + 15)
We have a= 1 ; b= 8 , c= 15
h= x_(vertex)= -8/(2*2) = color(red)-4
k= y_(vertex)= y(-4) = (-4)^2+8(-4)+15
y(-4) = 16-32+15 =color(red) (-1)
vertex form is color(blue)(y= 1(x-(-4))^2 + (-1))
simplify color(red)(y= 1(x+4))^color(red)2-1
