Subtract `3y^(2) - 2xy - (7) /(2) x^(2) " from " (-2)/(3) y^(2) - (3)/(2) xy + 4x^(2)`.

Options

A. `(11)/(3)y^(2) - (1)/(2) xy - (11)/(2) x^(2)`
B. `(11)/(3) y^(2) + (7)/(2) xy + (11)/(2) x^(2)`
C. `(-11)/(3) y^(2) + (1)/(2) xy + (15)/(2) x^(2)`
D. `(-11)/(3) y^(2) + (7)/(2) xy + (15)/(2)x^(2)`

Answer

To solve the problem of subtracting \(3y^2 - 2xy - \frac{7}{2}x^2\) from \(-\frac{2}{3}y^2 - \frac{3}{2}xy + 4x^2\), we will follow these steps:

### Step 1: Write down the expressions
Let:
- \(a = -\frac{2}{3}y^2 - \frac{3}{2}xy + 4x^2\)
- \(b = 3y^2 - 2xy - \frac{7}{2}x^2\)

We need to find \(a - b\).

### Step 2: Substitute the expressions into the subtraction
Now, we can substitute the expressions for \(a\) and \(b\) into the equation:
\[ a - b = \left(-\frac{2}{3}y^2 - \frac{3}{2}xy + 4x^2\right) - \left(3y^2 - 2xy - \frac{7}{2}x^2\right) \]

### Step 3: Distribute the negative sign
Distributing the negative sign across \(b\):
\[ a - b = -\frac{2}{3}y^2 - \frac{3}{2}xy + 4x^2 - 3y^2 + 2xy + \frac{7}{2}x^2 \]

### Step 4: Combine like terms
Now, we will combine like terms:

1. **For \(y^2\) terms**:
\[ -\frac{2}{3}y^2 - 3y^2 = -\frac{2}{3}y^2 - \frac{9}{3}y^2 = -\frac{11}{3}y^2 \]

2. **For \(xy\) terms**:
\[ -\frac{3}{2}xy + 2xy = -\frac{3}{2}xy + \frac{4}{2}xy = \frac{1}{2}xy \]

3. **For \(x^2\) terms**:
\[ 4x^2 + \frac{7}{2}x^2 = \frac{8}{2}x^2 + \frac{7}{2}x^2 = \frac{15}{2}x^2 \]

### Step 5: Write the final expression
Combining all the results from the previous step, we get:
\[ a - b = -\frac{11}{3}y^2 + \frac{1}{2}xy + \frac{15}{2}x^2 \]

### Final Answer:
\[ -\frac{11}{3}y^2 + \frac{1}{2}xy + \frac{15}{2}x^2 \]

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