The operation `square` is defined as `a square b=(2b^(2)-8a^(2))/(b+2a)` where a and b are real numbers and `b ne -2a.` What is the value of `(-2)square(-1)`?

Answer

To find the value of \((-2) \text{ square } (-1)\) using the defined operation \(a \text{ square } b = \frac{2b^2 - 8a^2}{b + 2a}\), we will follow these steps:

### Step 1: Identify the values of \(a\) and \(b\)
We have:
- \(a = -2\)
- \(b = -1\)

### Step 2: Substitute \(a\) and \(b\) into the operation
Now, we will substitute \(a\) and \(b\) into the expression:
\[ (-2) \text{ square } (-1) = \frac{2(-1)^2 - 8(-2)^2}{-1 + 2(-2)} \]

### Step 3: Calculate \(2(-1)^2\)
Calculating \(2(-1)^2\):
\[ (-1)^2 = 1 \quad \Rightarrow \quad 2 \times 1 = 2 \]

### Step 4: Calculate \(-8(-2)^2\)
Calculating \(-8(-2)^2\):
\[ (-2)^2 = 4 \quad \Rightarrow \quad -8 \times 4 = -32 \]

### Step 5: Combine the results in the numerator
Now, substituting back into the numerator:
\[ 2 - 32 = -30 \]

### Step 6: Calculate the denominator
Now, calculate the denominator:
\[ -1 + 2(-2) = -1 - 4 = -5 \]

### Step 7: Combine the numerator and denominator
Now we have:
\[ (-2) \text{ square } (-1) = \frac{-30}{-5} \]

### Step 8: Simplify the fraction
Simplifying \(\frac{-30}{-5}\):
\[ \frac{-30}{-5} = 6 \]

### Final Answer
Thus, the value of \((-2) \text{ square } (-1)\) is \(6\).

---