The given problem is to find the solution to the quadratic equation [tex]\(x^2 + 14x + 45 = 0\)[/tex] and identify any mistakes made in the steps provided by the student.
Let's go through each step to determine where the mistake occurred:
1. Step 1: Factoring the polynomial
[tex]\[
x^2 + 14x + 45 = (x + 5)(x + 9)
\][/tex]
To verify this factoring, we expand [tex]\((x + 5)(x + 9)\)[/tex]:
[tex]\[
(x + 5)(x + 9) = x^2 + 9x + 5x + 45 = x^2 + 14x + 45
\][/tex]
Factoring is correctly done in Step 1.
2. Step 2: Setting each factor to zero
The student sets each factor to zero to find the roots:
[tex]\[
x + 5 = 0 \quad \text{or} \quad x + 9 = 0
\][/tex]
It should be [tex]\(\textbf{x + 5 = 0}\)[/tex] or [tex]\(\textbf{x + 9 = 0}\)[/tex], not [tex]\(x + 5 = 0\)[/tex] or [tex]\(x - 9 = 0\)[/tex] as stated in the student's step. There is no equation [tex]\(x - 9 = 0\)[/tex] from the given factorization. Thus, the error is in Step 2.
3. Step 3: Solving the equations
If using [tex]\(x + 5 = 0\)[/tex] and [tex]\(x + 9 = 0\)[/tex]:
[tex]\[
x + 5 = 0 \implies x = -5
\][/tex]
[tex]\[
x + 9 = 0 \implies x = -9
\][/tex]
So, the correct solutions are [tex]\(x = -5\)[/tex] or [tex]\(x = -9\)[/tex].
Since the error is in step 2 where there is a mention of [tex]\(x - 9 = 0\)[/tex] instead of [tex]\(x + 9 = 0\)[/tex], the mistake lies here.
Therefore, the correct answer is:
[tex]\[ \boxed{D. \text{in Step 2}} \][/tex]
