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