Find the sum of the expressions
`- 3a ^(2) + 2 ab - 4b^(2) and - 6a ^(2) + 4b.`

Options

A. `- 9a ^(2) - 4b ^(2) + 2 ab + 4b`
B. `9a ^(2) + 4b ^(2) + 6 ab`
C. `- 9 a ^(2) - 4b ^(2) - 2 ab - 4 b`
D. None of the above

Answer

To find the sum of the expressions \(-3a^2 + 2ab - 4b^2\) and \(-6a^2 + 4b\), we will follow these steps:

### Step 1: Write down the expressions
We have two expressions:
1. \(-3a^2 + 2ab - 4b^2\)
2. \(-6a^2 + 4b\)

### Step 2: Combine like terms
We will add the coefficients of like terms together.

- For \(a^2\) terms:
\(-3a^2\) and \(-6a^2\)

\[ -3a^2 - 6a^2 = -9a^2 \]

- For \(ab\) terms:
We only have \(2ab\) from the first expression, and there is no \(ab\) term in the second expression.

\[ 2ab \]

- For \(b^2\) terms:
We have \(-4b^2\) from the first expression and no \(b^2\) term in the second expression.

\[ -4b^2 \]

- For \(b\) terms:
We have \(0\) from the first expression and \(4b\) from the second expression.

\[ 4b \]

### Step 3: Combine all the results
Now, we can combine all the results from the previous step:

\[ -9a^2 + 2ab - 4b^2 + 4b \]

### Final Expression
Thus, the sum of the expressions is:

\[ -9a^2 + 2ab - 4b^2 + 4b \]