Q:

The coefficent of x

I was wondering if this question is really teasing me. 

$f(x)=x^2+bx+c$, where $b=-17$ and $c=52$. What is the coefficient of $x^2$ in the function $f(f(x))$? 

So basically the coefficent will be 1? 

A:

$f(f(x)) = f(x)^2 + b f(x) + c = (x^2 + bx + c)^2 + b (x^2 + bx + c) + c$.
Now $(x^2 + bx + c)^2 = (x^2 + bx+c)(x^2 + bx + c) =$
$x^4 + 2bx^3 + (2c+b^2)x^2 + 2 b c x + c^2$.
In total, $x^2$ thus has the coefficient $2c+b^2 + b$, where the last $b$ comes from the linear term in the first line.