Question

Could someone help me understand what I need to do in this problem?

Answer 1

`p(x-1)` is a quadratic in the form

`a(x-1)^2 + b(x-1) + c`

We want to write
`p(x-1)` instead in the form

`a' x^2 + b' x + c`

To do this, simply expand the first form completely.

`a(x-1)^2 + b(x-1) + c = a(x^2 - 2x + 1) + b(x-1) + c \\\\ ~~~~~~~~ = ax^2 + (-2a + b)x + (a - b + c)`

Then
`a'=a`,
`b'=-2a+b`, and
`c'=a-b+c`.

Given

`p(x-1) = 2(x-1)^2 + 3(x-1) + 4`

(so
`a=2`,
`b=3`, and
`c=4`) it follows that

`a' = a = 2`

`b' = -2a + b = -2^2 + 3 = -1`

`c' = a - b + c = 2 - 3 + 4 = 3`

`\implies p(x-1) = \boxed{2}\,x^2 - \boxed{1}\,x + \boxed{3}`