Fill in the blank to show what the quadratic expression would look like when factored. (5 points)

Let's factor the given quadratic expressions step by step.

1. **For the expression** \( x^2 - 3x - 10 \):
- We need to find two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of the x term).
- The numbers \(-5\) and \(2\) fit this requirement because:
- \(-5 \times 2 = -10\)
- \(-5 + 2 = -3\)
- Therefore, we can express it as:
\[
x^2 - 3x - 10 = (x - 5)(x + 2)
\]

2. **For the expression** \( 3x^2 - 6x + 9 \):
- First, notice that all terms share a common factor of \(3\):
\[
3(x^2 - 2x + 3)
\]
- The expression \(x^2 - 2x + 3\) does not factor nicely into rational numbers because the discriminant (\(-2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8\)) is negative, indicating that it has complex roots. So, it does not have a real factorization over the integers.
- Therefore, the expression remains:
\[
3x^2 - 6x + 9 = 3(x^2 - 2x + 3)
\]

3. **For the expression** \( 9x^2 + 225 \):
- First, note that this is a sum of squares, which does not factor into real numbers. However, it can be simplified by factoring out a common term:
\[
9(x^2 + 25)
\]
- The expression \(x^2 + 25\) also cannot be factored over the reals. So, this is the simplest form you can get.
- The final result is:
\[
9x^2 + 225 = 9(x^2 + 25)
\]

So, to summarize:
1. \( x^2 - 3x - 10 = (x - 5)(x + 2) \)
2. \( 3x^2 - 6x + 9 = 3(x^2 - 2x + 3) \)
3. \( 9x^2 + 225 = 9(x^2 + 25) \)