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#