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------------------------------------------------ Select the correct answer.

Which of these is the standard form of the following function?

A. [tex]f(x) = -9(x+5)^2 + 4[/tex]

B. [tex]f(x) = 9x^2 - 180x + 221[/tex]

C. [tex]f(x) = 9x^2 - 90x - 221[/tex]

D. [tex]f(x) = -9x^2 - 90x - 221[/tex]

E. [tex]f(x) = -9x^2 - 180x - 221[/tex]

Let's find the standard form of the function [tex]$ f(x) = -9(x+5)^2 + 4 $[/tex].

1. Expand the Squared Term:

[tex]$(x + 5)^2$[/tex] can be expanded using the formula [tex]$(a + b)^2 = a^2 + 2ab + b^2$[/tex].

[tex]\[
(x + 5)^2 = x^2 + 10x + 25
\][/tex]

2. Distribute the Coefficient:

Now, multiply each term in the expression by [tex]$-9$[/tex]:

[tex]\[
-9(x^2 + 10x + 25) = -9x^2 - 90x - 225
\][/tex]

3. Add the Constant Term:

Finally, add the constant term [tex]$4$[/tex] to the expression:

[tex]\[
-9x^2 - 90x - 225 + 4 = -9x^2 - 90x - 221
\][/tex]

So, the standard form of the function is:

[tex]\[ f(x) = -9x^2 - 90x - 221 \][/tex]

The correct answer is the third option:
[tex]$ f(x) = -9x^2 - 90x - 221 $[/tex]