Factor completely. \( 5 w-10 w^{2}+2 w^{4}-w^{3} \)

To factor the expression \( 5w - 10w^2 + 2w^4 - w^3 \), we can first rearrange the terms in descending order of powers: \( 2w^4 - w^3 - 10w^2 + 5w \). Now, we can factor by grouping. Let's group the first two terms and the last two terms: \( (2w^4 - w^3) + (-10w^2 + 5w) \). From the first group, we can factor out \( w^3 \): \( w^3(2w - 1) \). From the second group, we can factor out \(-5w\): \( -5w(2w - 1) \). Now, we have: \( w^3(2w - 1) - 5w(2w - 1) \). We can factor out the common factor \( (2w - 1) \): \( (2w - 1)(w^3 - 5w) \). Next, the second factor \( w^3 - 5w \) can be factored further by taking out \( w \): \( w(w^2 - 5) \). Putting it all together, we have: \( (2w - 1)w(w^2 - 5) \). So the completely factored form of \( 5w - 10w^2 + 2w^4 - w^3 \) is: \[ (2w - 1)w(w^2 - 5). \]