The sum of `−2x^2 + x + 31` and `3x^2 + 7x − 8` can be written in the form `ax^2 + bx + c`, where a, b, and c are constants.What is the value of a + b + c ?

Explanation

To find the value of \( a + b + c \) from the sum of the polynomials \( -2x^2 + x + 31 \) and \( 3x^2 + 7x - 8 \), we will follow these steps:

### Step 1: Write down the two polynomials
We have:
1. \( P_1(x) = -2x^2 + x + 31 \)
2. \( P_2(x) = 3x^2 + 7x - 8 \)

### Step 2: Add the two polynomials
To find the sum \( P_1(x) + P_2(x) \), we combine like terms:
\[ P_1(x) + P_2(x) = (-2x^2 + 3x^2) + (x + 7x) + (31 - 8) \]

### Step 3: Combine the coefficients of \( x^2 \)
The coefficient of \( x^2 \):
\[ -2 + 3 = 1 \]

### Step 4: Combine the coefficients of \( x \)
The coefficient of \( x \):
\[ 1 + 7 = 8 \]

### Step 5: Combine the constant terms
The constant term:
\[ 31 - 8 = 23 \]

### Step 6: Write the resulting polynomial
Now we can write the resulting polynomial in the form \( ax^2 + bx + c \):
\[ P(x) = 1x^2 + 8x + 23 \]
Here, \( a = 1 \), \( b = 8 \), and \( c = 23 \).

### Step 7: Calculate \( a + b + c \)
Now we find:
\[ a + b + c = 1 + 8 + 23 = 32 \]

Thus, the final answer is:
\[ \boxed{32} \]