`x^(4)+4x^(3)+ax^(2)+bx+25` is a perfect square number, where a and b are positive real numbers then find the value of a + b.

Options

A. 20
B. 32
C. 34
D. 42

Answer

To determine the values of \( a \) and \( b \) in the expression \( x^4 + 4x^3 + ax^2 + bx + 25 \) such that it is a perfect square, we can follow these steps:

### Step 1: Assume the Perfect Square Form
We assume that the expression can be represented as the square of a quadratic polynomial:
\[ (x^2 + px + 5)^2 \]

### Step 2: Expand the Assumed Form
Now, we expand the assumed perfect square:
\[ (x^2 + px + 5)^2 = (x^2)^2 + 2(px)(x^2) + (px)^2 + 2(5)(x^2) + (5)^2 \]
This simplifies to:
\[ x^4 + 2px^3 + (p^2 + 10)x^2 + 10px + 25 \]

### Step 3: Compare Coefficients
Now we compare coefficients from the original polynomial \( x^4 + 4x^3 + ax^2 + bx + 25 \) with the expanded form:
- Coefficient of \( x^3 \): \( 2p = 4 \)
- Coefficient of \( x^2 \): \( p^2 + 10 = a \)
- Coefficient of \( x \): \( 10p = b \)

### Step 4: Solve for \( p \)
From \( 2p = 4 \):
\[ p = 2 \]

### Step 5: Calculate \( a \)
Substituting \( p = 2 \) into \( p^2 + 10 = a \):
\[ a = 2^2 + 10 = 4 + 10 = 14 \]

### Step 6: Calculate \( b \)
Substituting \( p = 2 \) into \( 10p = b \):
\[ b = 10 \times 2 = 20 \]

### Step 7: Find \( a + b \)
Now we find \( a + b \):
\[ a + b = 14 + 20 = 34 \]

### Final Answer
Thus, the value of \( a + b \) is:
\[ \boxed{34} \]