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The problem in the image asks to factor the polynomial \( 4x^2 - 12x + 16 \) completely.

Let's solve this step-by-step:

1. **Identify the common factor:**  
   The greatest common factor (GCF) among the terms is 4. So, first factor out 4:
   \[
   4(x^2 - 3x + 4)
   \]

2. **Check if the quadratic \(x^2 - 3x + 4\) can be factored further:**  
   To factor the quadratic expression \( x^2 - 3x + 4 \), we need to check the discriminant, \( \Delta \), of the quadratic formula. The discriminant is given by:
   \[
   \Delta = b^2 - 4ac
   \]
   where \( a = 1 \), \( b = -3 \), and \( c = 4 \). Substituting the values:
   \[
   \Delta = (-3)^2 - 4(1)(4) = 9 - 16 = -7
   \]
   Since the discriminant is negative (\( \Delta = -7 \)), the quadratic expression \( x^2 - 3x + 4 \) cannot be factored over the real numbers.

Therefore, the polynomial \( 4x^2 - 12x + 16 \) is already in its simplest factored form as \( 4(x^2 - 3x + 4) \), and it cannot be factored further.

Thus, the correct answer is **B**: "The polynomial is prime."

Would you like more details or have any questions? Here are some related questions for further practice:

1. How can you determine if a quadratic polynomial is factorable using the discriminant?
2. What does it mean when the discriminant of a quadratic is negative?
3. How do you factor out the greatest common factor (GCF) from a polynomial?
4. Can a polynomial with a negative discriminant be factored over complex numbers?
5. What is the general process for factoring a trinomial?

**Tip:** When factoring polynomials, always check for a greatest common factor first before attempting other factoring methods!

Factor the polynomial completely. If the polynomial cannot be factored, select 'The polynomial is prime'. The polynomial is: 4x^2 - 12x + 16

Learn how to factor the quadratic polynomial 4x^2 - 12x + 16 or determine if it is prime. This step-by-step guide covers key algebraic concepts, including factoring polynomials and using the discriminant formula to check for factorability. Suitable for students in Grades 8-10.

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