To factor the expression \(4x^4 - x^3 + 20x^2 - 5x\), we first notice that each term has a common factor of \(x\). We can factor out \(x\):
\[ x(4x^3 - x^2 + 20x - 5) \]
Next, we will need to factor the cubic polynomial \(4x^3 - x^2 + 20x - 5\). We can try to use the Rational Root Theorem or synthetic division to find possible roots, but factoring can also be done by trial and error with possible simple factors.
By trial, let’s test \(x = 1\):
\[ 4(1)^3 - (1)^2 + 20(1) - 5 = 4 - 1 + 20 - 5 = 18 \quad \text{(not a root)} \]
Testing \(x = -1\):
\[ 4(-1)^3 - (-1)^2 + 20(-1) - 5 = -4 - 1 - 20 - 5 = -30 \quad \text{(not a root)} \]
Testing \(x = \frac{1}{2}\):
\[ 4\left(\frac{1}{2}\right)^3 - \left(\frac{1}{2}\right)^2 + 20\left(\frac{1}{2}\right) - 5 = 4\left(\frac{1}{8}\right) - \left(\frac{1}{4}\right) + 10 - 5 \] \[ = \frac{1}{2} - \frac{1}{4} + 10 - 5 = \frac{1}{2} - \frac{1}{4} + 5 = \frac{1}{4} + 5 = \frac{21}{4} \quad \text{(not a root)} \]
After testing several roots, direct factoring seems challenging. However, we can apply polynomial division or synthetic division if we find possible factors.
Since factoring directly might be cumbersome, we can check for the given options. Given \(x^2 + 5\) or \(x^2 - 5\) or \(4x - 1\), let's test \(x(4x^3 - x^2 + 20x - 5) = 0\) or look for obvious factors.
If we systematically check for simple roots like \(x\), there would be quadratic factors of forms. If we denote that the higher degree term \(4x^4\) and other terms can be factor combinations of \(x\) with trial factors \(x^2\).
Through synthetic division or numerical methods, we need to find if \(4x^2 + 20 = 0\), and checking indicates if it's \(+5\) or \(-5\). In retrospect of options given \(x^2 + 5\) emerges closer.
Since the task involves identifying a given answer out of selected options from \(x^2 + 5, x^2 - 5, 4x - 1, x\):
The most likely answer from common polynomial formulations evident here, given works typically yield:
The quadratic factor might be \(x^2 + 5\) given typical factor choices, although it requires validating specially as it might vary based on domain or analysis. Thus while answers appear interlinked, identifying concludes returns \(x^2 + 5\) as the approachable solution within inputs.
Final answer: \(x^2 + 5\).
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