combining functions

[red and white kop!]

Functions f and g are defined by
f(x) = (x^2) + 2x + 3, x ϵ R
g(x) = ax + b, x ϵ R

Given that fg(x) = 4(x^2) – 48x + 146 for all x, find the possible values of a and b.

Alright so I’m a little lost here but I’m not looking for a straightforward answer, just any suggestions that could help me. Up until now I only had to deal with pretty simple problems involving combining functions, so this one has me confused.
This is my work up until now:

4(x^2) – 48x + 146 = (ax + b)^2 + 2(ax + b) + 3
4(x^2) – 48x + 146 = (a^2)(x^2) + 2abx + (b^2) +2ax + 2b + 3
=> (4 – (a^2))(x^2) – (2ab + 2a + 48)x + 143 – (b^2) – 2b = 0

So yes, all I’ve done is group the terms into quadratic form, and I have no idea where to go from here.
Helpful hints?

 

[srmichael]

You were on the right track up until the last line. The second line says:

$\displaystyle a^2x^2+(2ab)x+b^2+(2a)x+2b+3=4x^2-48x+146$

So let me ask you a question: If $\displaystyle ax^2+4x+10=9x^2+4x+10$, what does "a" equal? 9, right? So, see if you can use this same logic and apply it to the line above (hint: write the left hand side in the standard form of a quadratic function $\displaystyle ax^2+bx+c$

Hope this helps!

 

[red and white kop!]

:cool: