I think you want to separate a perfect square, i.e. express in the form (x-b)^2+a or -(x-b)^2+a.

As you know, (x-b)^2 = x^2 - 2*b*x + b^2, and -(x-b)^2 = -x^2 + 2*b*x - b^2. Original expression has +x^2, so only the first form is possible.

It is simple to find b, because -2*b = -8, so b=4. Then find a: (x-4)^2 + a = x^2 - 8x + 16 + a = x^2 - 8x - 3. Thus 16+a = -3, a=-19.

The answer: x^2-8x-3 = (x-4)^2-19.

Given

`x^2 - 8x -3` -------- ( 1)

to represent the above expression in the form of a(x-b)^2+c or a(x-b)^2-c

Expand `a((x-b)^2)+c`

= `a(x^2-2bx+b^2)+c`

= `a*(x^2)-2*a*b*x+(a*(b^2))+c-------------------(2)`

Now compare 1 and 2 equations we get,

**1)** a=1

**2)**

`2*a*b` = 8

=> as a= 1 so b= 4 on solving

3)

`a*(b^2)+c` = -3

we know the values of a,b on substituting

`1*(4^2)+c =-3`

`16+c =-3`

`c=-19`

Now from Equation 2 we get

`a((x-b)^2)+c` = `1*((x-4)^2)-19`

= `((x-4)^2)-19` is the form which is required