Sure, let's solve this step-by-step.
Given the function [tex]\( g(x) = 2x^2 + 4x + 6 \)[/tex], we need to find [tex]\( g(p+4) \)[/tex].
1. Substitute [tex]\( p+4 \)[/tex] for [tex]\( x \)[/tex] in the function [tex]\( g(x) \)[/tex]:
[tex]\[
g(p+4) = 2(p+4)^2 + 4(p+4) + 6
\][/tex]
2. Expand the squared term [tex]\( (p+4)^2 \)[/tex]:
[tex]\[
(p+4)^2 = p^2 + 8p + 16
\][/tex]
3. Substitute this expansion back into the expression:
[tex]\[
g(p+4) = 2(p^2 + 8p + 16) + 4(p+4) + 6
\][/tex]
4. Distribute the 2 through the first group of terms:
[tex]\[
g(p+4) = 2p^2 + 16p + 32 + 4(p+4) + 6
\][/tex]
5. Distribute the 4 through the second group of terms:
[tex]\[
g(p+4) = 2p^2 + 16p + 32 + 4p + 16 + 6
\][/tex]
6. Combine like terms:
[tex]\[
g(p+4) = 2p^2 + (16p + 4p) + (32 + 16 + 6)
\][/tex]
[tex]\[
g(p+4) = 2p^2 + 20p + 54
\][/tex]
So, the simplified expression for [tex]\( g(p+4) \)[/tex] is:
[tex]\[
g(p+4) = 2p^2 + 20p + 54
\][/tex]
