Answer :

Sure, let's rewrite the given function by completing the square.

1. Starting Function:
[tex]\[ f(x) = x^2 + 14x + 8 \][/tex]

2. Grouping the [tex]$x$[/tex] Terms:
To complete the square, we first look at the quadratic and linear terms: [tex]\(x^2 + 14x\)[/tex].

3. Forming a Perfect Square:
For the square term [tex]\((x + d)^2\)[/tex] to match [tex]\(x^2 + 14x\)[/tex], the constant term [tex]\(d\)[/tex] can be found using:
[tex]\[ d = \frac{b}{2a} = \frac{14}{2 \cdot 1} = 7 \][/tex]
Therefore, we want:
[tex]\[ (x + 7)^2 = x^2 + 14x + 49 \][/tex]

4. Adjusting the Constant Term:
Note that [tex]\((x + 7)^2\)[/tex] introduces an extra term [tex]\(49\)[/tex] that was not in the original equation, so we need to subtract this extra number:
[tex]\[ x^2 + 14x + 8 = (x + 7)^2 - 49 + 8 \][/tex]

5. Simplifying the Added/Subtracted Terms:
Now combine the constant terms:
[tex]\[ -49 + 8 = -41 \][/tex]

6. Final Function:
Therefore, we have:
[tex]\[ f(x) = (x + 7)^2 - 41 \][/tex]

So, the completed square form of the function is:
[tex]\[ f(x) = (x + 7)^2 - 41 \][/tex]