When the product $(3x+2y+1)(x+4y+5)$ is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of $y$?
We multiply out using the distributive property:

\begin{align*}
&\phantom{==}(3x+2y+1)(x+4y+5)\\
&=3x(x+4y+5)+2y(x+4y+5)+1(x+4y+5)\\
&=3x^2+12xy+15x+2xy+8y^2+10y+x+4y+5\\
&=3x^2+14xy+16x+8y^2+14y+5.
\end{align*}Those terms which contain some power of $y$ are $14xy$, $8y^2$, and $14y$, and the sum of the coefficients is $14+8+14=\boxed{36}$.
36
Level 5
Algebra
Problem: When the product $(3x+2y+1)(x+4y+5)$ is expanded, what is the sum of the coefficients of the terms which contain a nonzero power of $y$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the sum of the coefficients of the terms containing a nonzero power of \( y \) in the product \( (3x + 2y + 1)(x + 4y + 5) \), we first need to expand the expression. 

We can use the distributive property to expand the product. Each term in the first polynomial must be multiplied by each term in the second polynomial. The first polynomial has three terms: \( 3x \), \( 2y \), and \( 1 \). The second polynomial also has three terms: \( x \), \( 4y \), and \( 5 \).

We will calculate the contributions to the terms with \( y \) from each multiplication:

1. From \( 3x \):
   - \( 3x \cdot 4y = 12xy \)
   - \( 3x \cdot 5 = 15x \) (no contribution to \( y \))

2. From \( 2y \):
   - \( 2y \cdot x = 2xy \)
   - \( 2y \cdot 4y = 8y^2 \)
   - \( 2y \cdot 5 = 10y \)

3. From \( 1 \):
   - \( 1 \cdot 4y = 4y \)
   - \( 1 \cdot 5 = 5 \) (no contribution to \( y \))

Now, we collect all the terms that contain \( y \):
- From \( 3x \): \( 12xy \)
- From \( 2y \): \( 2xy + 8y^2 + 10y \)
- From \( 1 \): \( 4y \)

Combining these, we have:
- The coefficient of \( xy \) is \( 12 + 2 = 14 \).
- The coefficient of \( y^2 \) is \( 8 \).
- The coefficient of \( y \) is \( 10 + 4 = 14 \).

Next, we sum the coefficients of all terms that contain a nonzero power of \( y \):
- Coefficient of \( xy \): \( 14 \)
- Coefficient of \( y^2 \): \( 8 \)
- Coefficient of \( y \): \( 14 \)

Thus, the total sum of the coefficients is:
\[
14 + 8 + 14 = 36
\]

Therefore, the final answer is \( 36 \).<|END_OF_LATENT|> The final answer is $36$.