Subtract `5x^2 - 4y^2 + 6y - 3` from `7x^2 - 4xy + 8y^2 + 5x - 3y`.

Explanation

To solve the problem of subtracting the expression \(5x^2 - 4y^2 + 6y - 3\) from \(7x^2 - 4xy + 8y^2 + 5x - 3y\), we can follow these steps:

### Step-by-Step Solution:

1. **Write the expressions**: We have two expressions:
- Expression 1: \(7x^2 - 4xy + 8y^2 + 5x - 3y\)
- Expression 2: \(5x^2 - 4y^2 + 6y - 3\)

2. **Set up the subtraction**: We need to subtract Expression 2 from Expression 1. This can be written as:
\[ (7x^2 - 4xy + 8y^2 + 5x - 3y) - (5x^2 - 4y^2 + 6y - 3) \]

3. **Distribute the negative sign**: When we subtract the second expression, we need to distribute the negative sign across all terms in Expression 2:
\[ 7x^2 - 4xy + 8y^2 + 5x - 3y - 5x^2 + 4y^2 - 6y + 3 \]

4. **Combine like terms**:
- Combine \(x^2\) terms: \(7x^2 - 5x^2 = 2x^2\)
- Combine \(y^2\) terms: \(8y^2 + 4y^2 = 12y^2\)
- The \(xy\) term remains: \(-4xy\)
- Combine \(x\) terms: \(5x\) remains as it is.
- Combine \(y\) terms: \(-3y - 6y = -9y\)
- Combine constant terms: \(3\) remains as it is.

Putting it all together, we have:
\[ 2x^2 - 4xy + 12y^2 + 5x - 9y + 3 \]

5. **Final arrangement**: The final expression can be arranged as:
\[ 2x^2 + 12y^2 + 5x - 4xy - 9y + 3 \]

### Final Answer:
Thus, the result of subtracting \(5x^2 - 4y^2 + 6y - 3\) from \(7x^2 - 4xy + 8y^2 + 5x - 3y\) is:
\[ 2x^2 - 4xy + 12y^2 + 5x - 9y + 3 \]