Find he value of `a^(2)+ab^(2)+a^(2)b+b^(2)` at `a= -1` and b= 2.

Answer

To find the value of the expression \( a^2 + ab^2 + a^2b + b^2 \) at \( a = -1 \) and \( b = 2 \), we can follow these steps:

### Step 1: Substitute the values of \( a \) and \( b \) into the expression.
We have:
- \( a = -1 \)
- \( b = 2 \)

So, substituting these values into the expression gives us:
\[ (-1)^2 + (-1)(2^2) + (-1)^2(2) + (2^2) \]

### Step 2: Calculate each term in the expression.
1. Calculate \( (-1)^2 \):
\[ (-1)^2 = 1 \]

2. Calculate \( (-1)(2^2) \):
\[ (-1)(2^2) = (-1)(4) = -4 \]

3. Calculate \( (-1)^2(2) \):
\[ (-1)^2(2) = 1 \cdot 2 = 2 \]

4. Calculate \( (2^2) \):
\[ (2^2) = 4 \]

### Step 3: Combine all the calculated terms.
Now, we can combine all the results:
\[ 1 - 4 + 2 + 4 \]

### Step 4: Simplify the expression.
1. Start with \( 1 - 4 \):
\[ 1 - 4 = -3 \]

2. Now add \( 2 \):
\[ -3 + 2 = -1 \]

3. Finally, add \( 4 \):
\[ -1 + 4 = 3 \]

### Final Result:
The value of the expression \( a^2 + ab^2 + a^2b + b^2 \) at \( a = -1 \) and \( b = 2 \) is \( 3 \).

---