We are given the polynomial $p(x) = 3 + 4x + 5x^2$ and asked to find its coordinate with respect to the basis $B = \{1, x, x^2\}$.

Step-by-Step Solution:

1. The given basis is $B = \{1, x, x^2\}$, which means that any polynomial can be written as a linear combination of the elements of this basis.

2. The polynomial $p(x)$ is: $p(x) = 3 + 4x + 5x^2$ Notice that:

   • The coefficient of $1$ (which corresponds to the basis element $1$) is $3$,
   • The coefficient of $x$ (which corresponds to the basis element $x$) is $4$,
   • The coefficient of $x^2$ (which corresponds to the basis element $x^2$) is $5$.

3. The coordinates of $p(x)$ with respect to the basis $B = \{1, x, x^2\}$ are just the coefficients of the terms in the polynomial: $[p(x)]_B = (3, 4, 5)$

4. Looking at the provided choices:

   • $(4, 5)$
   • $(0, 4, 5)$
   • $(3, 4, 5)$

   The correct answer is $\boxed{(3, 4, 5)}$.

Would you like more details on any of the steps?

Here are some related questions for practice:

1. How do you find the coordinates of a polynomial with respect to a different basis?
2. What are the steps to determine a polynomial basis?
3. How can we check if a set of polynomials forms a basis?
4. How do we convert coordinates with respect to one basis to another?
5. Can we apply this method to find coordinates of non-polynomial functions?

Tip: Always ensure that the polynomial and the basis terms are ordered correctly to avoid mistakes when writing out the coordinates!