`25x^(2)-(x^(2)-36)^(2)`=_____

Options

A. `(x-4)(x+4)(x+9)(x-9)`
B. `(x-4)(4+x)(x+9)(9-x)`
C. `(x+4)(x+4)(x-9)(x-9)`
D. `(x-4)(4-x)(x+9)(9+x)`

Answer

To solve the equation \( 25x^2 - (x^2 - 36)^2 \), we will follow these steps:

### Step 1: Rewrite the expression
We start with the expression:
\[ 25x^2 - (x^2 - 36)^2 \]
We can recognize \( 25x^2 \) as \( (5x)^2 \) and rewrite the expression:
\[ (5x)^2 - (x^2 - 36)^2 \]

### Step 2: Apply the difference of squares formula
The expression is now in the form of \( a^2 - b^2 \), where \( a = 5x \) and \( b = (x^2 - 36) \). We can apply the difference of squares formula:
\[ a^2 - b^2 = (a - b)(a + b) \]
Thus, we have:
\[ (5x - (x^2 - 36))(5x + (x^2 - 36)) \]

### Step 3: Simplify the factors
Now we simplify the two factors:
1. For \( 5x - (x^2 - 36) \):
\[ 5x - x^2 + 36 = -x^2 + 5x + 36 \]
2. For \( 5x + (x^2 - 36) \):
\[ 5x + x^2 - 36 = x^2 + 5x - 36 \]

So, our expression becomes:
\[ (-x^2 + 5x + 36)(x^2 + 5x - 36) \]

### Step 4: Factor the quadratic expressions
Now we will factor each quadratic expression:
1. For \( -x^2 + 5x + 36 \), we can factor out a negative sign:
\[ -(x^2 - 5x - 36) \]
To factor \( x^2 - 5x - 36 \), we need two numbers that multiply to \(-36\) and add to \(-5\). These numbers are \(-9\) and \(4\):
\[ -(x - 9)(x + 4) \]

2. For \( x^2 + 5x - 36 \), we need two numbers that multiply to \(-36\) and add to \(5\). These numbers are \(9\) and \(-4\):
\[ (x + 9)(x - 4) \]

### Step 5: Combine the factors
Now we can combine the factors:
\[ -(x - 9)(x + 4)(x + 9)(x - 4) \]

### Final Answer
Thus, the final factorized form of the expression is:
\[ -(x - 9)(x + 4)(x + 9)(x - 4) \]