Answer :

To rewrite the function [tex]\( f(x) = x^2 + 20x + 40 \)[/tex] by completing the square, follow these steps:

1. Identify the coefficient of [tex]\(x\)[/tex]: For the quadratic term [tex]\(x^2\)[/tex], the coefficient of [tex]\(x\)[/tex] is 20.

2. Take half the coefficient of [tex]\(x\)[/tex]: Half of 20 is 10.

3. Square this value: The square of 10 is 100.

4. Rewrite the middle term: Add and subtract this square (100) within the function to complete the square.

[tex]\[ f(x) = x^2 + 20x + 40 \][/tex]
Add and subtract 100:
[tex]\[ f(x) = x^2 + 20x + 100 - 100 + 40 \][/tex]

5. Form the perfect square trinomial:
[tex]\[ f(x) = (x + 10)^2 - 60 \][/tex]

So the function [tex]\( f(x) \)[/tex] rewritten by completing the square is:
[tex]\[ f(x) = (x + 10)^2 - 60 \][/tex]