Factories:
`4y ^(2) + 20y + 25`

Answer

To factor the quadratic expression \(4y^2 + 20y + 25\), we will follow these steps:

### Step 1: Identify the coefficients
The given quadratic expression is in the form \(ay^2 + by + c\), where:
- \(a = 4\)
- \(b = 20\)
- \(c = 25\)

### Step 2: Calculate the product \(ac\)
We need to find the product of \(a\) and \(c\):
\[ ac = 4 \times 25 = 100 \]

### Step 3: Find two numbers that multiply to \(ac\) and add to \(b\)
We need to find two numbers that multiply to \(100\) (the value of \(ac\)) and add up to \(20\) (the value of \(b\)). The two numbers that satisfy these conditions are \(10\) and \(10\):
\[ 10 \times 10 = 100 \quad \text{and} \quad 10 + 10 = 20 \]

### Step 4: Rewrite the middle term using the two numbers
Now we can rewrite the expression by splitting the middle term \(20y\) into \(10y + 10y\):
\[ 4y^2 + 10y + 10y + 25 \]

### Step 5: Factor by grouping
Next, we will group the terms:
\[ (4y^2 + 10y) + (10y + 25) \]
Now, we factor out the common factors from each group:
- From the first group \(4y^2 + 10y\), we can factor out \(2y\):
\[ 2y(2y + 5) \]
- From the second group \(10y + 25\), we can factor out \(5\):
\[ 5(2y + 5) \]

### Step 6: Combine the factors
Now we can combine the factored groups:
\[ 2y(2y + 5) + 5(2y + 5) = (2y + 5)(2y + 5) \]
This can also be written as:
\[ (2y + 5)^2 \]

### Final Answer
Thus, the factorization of the expression \(4y^2 + 20y + 25\) is:
\[ (2y + 5)^2 \]

---