Functions

Given two functions, f(x)=ax2+bx+c, which is even, and g(x)=px3+qx2+rx, which is odd, we need to analyze the combined function h(x)=f(x)+g(x). We know that h(−2)=0.
Properties of Even and Odd Functions:
Even Function: f(x)=f(−x).
Odd Function: g(x)=−g(−x).
Since f(x) is even:
f(x)=f(−x)‌‌⇒‌‌f(2)=f(−2)
For the odd function g(x) :
g(−x)=−g(x)‌‌⇒‌‌g(−2)=−g(2)
Using the Given Information:
Given h(−2)=0, we have:
h(−2)=f(−2)+g(−2)=0
From the properties mentioned:
f(−2)=f(2)‌‌‌ and ‌‌‌g(−2)=−g(2)
Substitute these into h(−2)=0 :
f(2)−g(2)=0

This implies:
f(2)=g(2)
Expressing f(2) and g(2) :
Calculate f(2) :
f(2)=a(2)2+b(2)+c=4a+2b+c
And g(2) :
g(2)=p(2)3+q(2)2+r(2)=8p+4q+2r
Equating the two expressions from f(2)=g(2) :
4a+2b+c=8p+4q+2r
Thus, the expression for 8p+4q+2r is:
8p+4q+2r=4a+2b+c