To find the span of the set of vectors \( \{1 + 4x + x^2, 2 + 5x + x^2\} \), we need to determine all possible linear combinations of these two vectors.

Let \( f_1(x) = 1 + 4x + x^2 \) and \( f_2(x) = 2 + 5x + x^2 \). A general linear combination of these two vectors is:

\[
\alpha(1 + 4x + x^2) + \beta(2 + 5x + x^2) = (\alpha + 2\beta) + (4\alpha + 5\beta)x + (\alpha + \beta)x^2
\]

This expression shows that any function in the span of \( f_1(x) \) and \( f_2(x) \) is of the form:

\[
(\alpha + 2\beta) + (4\alpha + 5\beta)x + (\alpha + \beta)x^2
\]

Now, let's analyze the coefficients:

- Constant term: \( \alpha + 2\beta \)
- Coefficient of \( x \): \( 4\alpha + 5\beta \)
- Coefficient of \( x^2 \): \( \alpha + \beta \)

The span of \( \{1 + 4x + x^2, 2 + 5x + x^2\} \) consists of all quadratic polynomials of the form:

\[
c_0 + c_1x + c_2x^2
\]

where \( c_0 \), \( c_1 \), and \( c_2 \) are real numbers that can be achieved by appropriate choices of \( \alpha \) and \( \beta \).

### Conclusion:
The span of \( \{1 + 4x + x^2, 2 + 5x + x^2\} \) is the space of all quadratic polynomials, i.e., \( \text{Span}\{1 + 4x + x^2, 2 + 5x + x^2\} = \mathbb{P}_2 \), which is the set of all polynomials of degree at most 2.

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### Related Questions:
1. How do you determine the dimension of a span?
2. What is the general form of the span of three quadratic polynomials?
3. How can you find a basis for the space spanned by a set of vectors?
4. What is the definition of a vector space in more formal terms?
5. How do linear combinations help in understanding vector spaces?

### Tip:
Whenever you're finding the span of functions or vectors, focus on combining them using coefficients to see the structure of the resulting space.

Explore how to find the span of {1 + 4x + x^2, 2 + 5x + x^2}. This problem introduces key concepts from linear algebra, such as vector spaces, span, and quadratic polynomials. Suitable for undergraduate students studying linear algebra.

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