Factor completely: $5u^5 - 10u^4 - 4u^3 + 8u^2$

Steps to Solve

1. Find the Greatest Common Factor (GCF)

Find the GCF of the coefficients: 5, 10, 4, and 8. The GCF is 1.

Find the GCF of the variable terms: $u^5$, $u^4$, $u^3$, and $u^2$. The GCF is $u^2$.

Therefore, the GCF of the entire polynomial is $1 \cdot u^2 = u^2$.

2. Factor out the GCF

Factor $u^2$ from the polynomial $5u^5 - 10u^4 - 4u^3 + 8u^2$: $5u^5 - 10u^4 - 4u^3 + 8u^2 = u^2(5u^3 - 10u^2 - 4u + 8)$

3. Factor by Grouping

Factor the expression inside the parenthesis, $5u^3 - 10u^2 - 4u + 8$, by grouping. Group the first two terms and the last two terms: $(5u^3 - 10u^2) + (-4u + 8)$

Factor out the GCF from each group: $5u^2(u - 2) - 4(u - 2)$

Now, factor out the common binomial factor $(u - 2)$: $(u - 2)(5u^2 - 4)$

4. Factor the Difference of Squares

Notice that $(5u^2 - 4)$ can be written as $(\sqrt{5}u)^2 - (2)^2$, which is a difference of squares. Use the difference of squares formula $a^2 - b^2 = (a - b)(a + b)$ to factor $(\sqrt{5}u)^2 - (2)^2$: $5u^2 - 4 = (\sqrt{5}u - 2)(\sqrt{5}u + 2)$

5. Write the Complete Factorization

Combine all the factors we found: $u^2(u - 2)(5u^2 - 4) = u^2(u - 2)(\sqrt{5}u - 2)(\sqrt{5}u + 2)$