To rewrite the given function [tex]\( f(x) = 4(x + 6)^2 + 5 \)[/tex] in its standard form, we need to expand it and rewrite it in the form [tex]\( f(x) = ax^2 + bx + c \)[/tex].
1. Start with the given function:
[tex]\[
f(x) = 4(x + 6)^2 + 5
\][/tex]
2. Expand [tex]\( (x + 6)^2 \)[/tex]:
[tex]\[
(x + 6)^2 = x^2 + 12x + 36
\][/tex]
3. Substitute this expansion back into the function:
[tex]\[
f(x) = 4(x^2 + 12x + 36) + 5
\][/tex]
4. Distribute the 4 across the terms inside the parentheses:
[tex]\[
f(x) = 4x^2 + 48x + 144 + 5
\][/tex]
5. Combine the constants 144 and 5:
[tex]\[
f(x) = 4x^2 + 48x + 149
\][/tex]
Therefore, the standard form of the function is:
[tex]\[
f(x) = 4x^2 + 48x + 149
\][/tex]
The values of the coefficients are:
[tex]\[
a = 4, \quad b = 48, \quad c = 149
\][/tex]
