Find the excess of `4p^(2)-2pq +3q^(2)` over `2p^(2)-pq +4q^(2)`.

Answer

To find the excess of \( 4p^2 - 2pq + 3q^2 \) over \( 2p^2 - pq + 4q^2 \), we need to subtract the second expression from the first. Here are the steps to solve the problem:

### Step 1: Write down the expressions
We have two algebraic expressions:
1. \( A = 4p^2 - 2pq + 3q^2 \)
2. \( B = 2p^2 - pq + 4q^2 \)

### Step 2: Set up the subtraction
We need to find the excess, which means we will subtract \( B \) from \( A \):
\[ \text{Excess} = A - B = (4p^2 - 2pq + 3q^2) - (2p^2 - pq + 4q^2) \]

### Step 3: Distribute the negative sign
Distributing the negative sign across the second expression:
\[ = 4p^2 - 2pq + 3q^2 - 2p^2 + pq - 4q^2 \]

### Step 4: Combine like terms
Now, we will combine the like terms:
- For \( p^2 \) terms: \( 4p^2 - 2p^2 = 2p^2 \)
- For \( pq \) terms: \( -2pq + pq = -pq \)
- For \( q^2 \) terms: \( 3q^2 - 4q^2 = -q^2 \)

Putting it all together, we get:
\[ \text{Excess} = 2p^2 - pq - q^2 \]

### Final Answer
The excess of \( 4p^2 - 2pq + 3q^2 \) over \( 2p^2 - pq + 4q^2 \) is:
\[ \boxed{2p^2 - pq - q^2} \]