Add :
`4x^(2)y, - 3xy^(2), -5xy^(2), 5x^(2)y`

Answer

To solve the problem of adding the algebraic expressions \(4x^2y\), \(-3xy^2\), \(-5xy^2\), and \(5x^2y\), we will follow these steps:

### Step 1: Identify Like Terms
First, we need to identify the like terms in the given expressions. The like terms are those that have the same variables raised to the same powers.

- The terms \(4x^2y\) and \(5x^2y\) are like terms because they both contain \(x^2y\).
- The terms \(-3xy^2\) and \(-5xy^2\) are like terms because they both contain \(xy^2\).

### Step 2: Group the Like Terms
Now, we will group the like terms together:

\[ (4x^2y + 5x^2y) + (-3xy^2 - 5xy^2) \]

### Step 3: Add the Like Terms
Next, we will add the coefficients of the like terms:

1. For \(x^2y\):
\[ 4 + 5 = 9 \]
So, \(4x^2y + 5x^2y = 9x^2y\).

2. For \(xy^2\):
\[ -3 + (-5) = -8 \]
So, \(-3xy^2 - 5xy^2 = -8xy^2\).

### Step 4: Combine the Results
Now we combine the results from the addition of like terms:

\[ 9x^2y - 8xy^2 \]

### Final Answer
The final answer is:

\[ 9x^2y - 8xy^2 \]