To add the polynomial expressions \( 3y - 4y^2 + 7y + 2 \) and \( 5y^2 + 6 - 8y \), we first combine like terms.
1. **Identify and arrange the terms:**
- From the first polynomial:
- \( -4y^2 \) (the \( y^2 \) term)
- \( 3y + 7y = 10y \) (the \( y \) terms)
- \( 2 \) (the constant term)
- From the second polynomial:
- \( 5y^2 \) (the \( y^2 \) term)
- \( -8y \) (the \( y \) term)
- \( 6 \) (the constant term)
2. **Add the like terms:**
- For \( y^2 \) terms:
\[
-4y^2 + 5y^2 = 1y^2 \text{ or simply } y^2
\]
- For \( y \) terms:
\[
10y - 8y = 2y
\]
- For the constant terms:
\[
2 + 6 = 8
\]
3. **Combine all the results:**
The sum of the polynomials is:
\[
y^2 + 2y + 8
\]
Thus, the final result is:
\[
\boxed{y^2 + 2y + 8}
\]