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The given expression is:

\[
25x^4 - 109x^2 + 36
\]

This is a quadratic in terms of \( x^2 \). We can treat it as a quadratic equation by letting \( y = x^2 \). The expression becomes:

\[
25y^2 - 109y + 36
\]

### Step 1: Factor the quadratic equation
We need to find two numbers whose product is \( 25 \times 36 = 900 \) and whose sum is \( -109 \). These numbers are \( -100 \) and \( -9 \), since:

\[
-100 \times -9 = 900 \quad \text{and} \quad -100 + (-9) = -109
\]

### Step 2: Rewrite and factor by grouping
We rewrite the middle term:

\[
25y^2 - 100y - 9y + 36
\]

Now, we factor by grouping:

\[
(25y^2 - 100y) - (9y - 36)
\]

Factor out the common terms:

\[
25y(y - 4) - 9(y - 4)
\]

Now, factor out \( (y - 4) \):

\[
(25y - 9)(y - 4)
\]

### Step 3: Substitute back \( y = x^2 \)
Substitute \( y = x^2 \) back into the factored form:

\[
(25x^2 - 9)(x^2 - 4)
\]

### Step 4: Further factor both terms
Both \( 25x^2 - 9 \) and \( x^2 - 4 \) are difference of squares, so we can factor them further:

\[
25x^2 - 9 = (5x - 3)(5x + 3)
\]
\[
x^2 - 4 = (x - 2)(x + 2)
\]

Thus, the full factorization is:

\[
(5x - 3)(5x + 3)(x - 2)(x + 2)
\]

### Step 5: Identify the linear factors
The linear factors are \( 5x - 3 \), \( 5x + 3 \), \( x - 2 \), and \( x + 2 \).

### Step 6: Sum of linear factors
The sum of the linear factors is:

\[
(5x - 3) + (5x + 3) + (x - 2) + (x + 2)
\]

Simplifying:

\[
5x + 5x + x + x = 12x
\]
\[
-3 + 3 - 2 + 2 = 0
\]

So, the sum of the linear factors is:

\[
12x
\]

### Final Answer:
The correct answer is **A. 12x**.

---

Do you want more details or have any questions? Here are some related questions you might find useful:

1. How do you factor a quadratic equation by grouping?
2. What is the difference of squares, and how is it factored?
3. How do you identify the sum of linear factors in a factored expression?
4. Can you apply substitution to factor higher-degree polynomials?
5. What are some other methods for factoring polynomials?

**Tip:** Always look for special patterns like the difference of squares, which can simplify complex factoring problems!

Factor the expression 25x^4 - 109x^2 + 36 and find the sum of the linear factors.

Learn how to factor the expression 25x^4 - 109x^2 + 36, applying factoring techniques and the difference of squares. The sum of the linear factors is calculated for Grades 9-12 math students.

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