To find the Greatest Common Factor (GCF) of the polynomial \(4x^4 + 12x^3 - 36x^2 + 24x\), we start by identifying the coefficients and the variable components of each term.
- The coefficients are: 4, 12, -36, 24.
- The variable parts are \(x^4, x^3, x^2, x\).
Step 1: Find the GCF of the coefficients.
- The GCF of the coefficients (4, 12, -36, 24) is 12.
Step 2: Find the lowest power of the variable \(x\).
- The lowest power of \(x\) present is \(x^1\).
Step 3: Combine these results.
- Putting the GCF of the coefficients and the lowest power of \(x\) together, we find that the GCF is \(12x\).
However, it looks like \(12x\) is not one of the options provided. Therefore, we should check if there is a simplification among the options given.
Let's check the options:
- 4
- \(4\)
- \(2x\)
- \(2\)
- \(6x\)
- \(6\)
- \(4x\)
Among these options, the highest expression that divides the original polynomial completely when simplified is \(4x\).
Thus, the GCF of the polynomial \(4x^4 + 12x^3 - 36x^2 + 24x\) is:
Final answer: \(4x\)