Suppose $d\not=0$.  We can write $\left(12d+13+14d^2\right)+\left(2d+1\right)$, in the form $ad+b+cd^2$, where $a$, $b$, and $c$ are integers.  Find $a+b+c$.
Adding the $d$ terms gives us $14d$.  Adding the constant terms gives us $14$.  Adding the $d^2$ terms gives us $14d^2$.  Adding the terms together gives us ${14d+14+14d^2}$, so $a+b+c = \boxed{42}$.
42
Level 2
Algebra
Problem: Suppose $d\not=0$.  We can write $\left(12d+13+14d^2\right)+\left(2d+1\right)$, in the form $ad+b+cd^2$, where $a$, $b$, and $c$ are integers.  Find $a+b+c$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the values of \(a\), \(b\), and \(c\) in the expression \(ad + b + cd^2\), we first need to simplify the given expression \((12d + 13 + 14d^2) + (2d + 1)\). 

Start by combining like terms:

1. Identify the terms involving \(d^2\):
   - From \(14d^2\), we have \(c = 14\).

2. Identify the terms involving \(d\):
   - From \(12d\) and \(2d\), we combine them: \(12d + 2d = 14d\). Thus, \(a = 14\).

3. Identify the constant terms:
   - From \(13\) and \(1\), we combine them: \(13 + 1 = 14\). Therefore, \(b = 14\).

Now we have determined:
- \(a = 14\)
- \(b = 14\)
- \(c = 14\)

Next, we calculate \(a + b + c\):
\[
a + b + c = 14 + 14 + 14 = 42.
\]

Thus, the final answer is \(42\).<|END_OF_LATENT|> The final answer is $42$.