To factor the given polynomial $2u^3 + 12u^2 - u - 6$ by grouping, follow these steps:

1. Group terms: $(2u^3 + 12u^2) - (u + 6)$

2. Factor out the greatest common factor (GCF) from each group:

   • From the first group $2u^3 + 12u^2$, factor out $2u^2$: $2u^2(u + 6)$
   • From the second group $- (u + 6)$, factor out $-1$: $-1(u + 6)$

3. Factor out the common binomial factor $(u + 6)$: $(2u^2 - 1)(u + 6)$

4. Check for further factorization:

   • Notice that $2u^2 - 1$ is a difference of squares: $2u^2 - 1 = ( \sqrt{2}u - 1)(\sqrt{2}u + 1)$
   • But if you stick to integer coefficients, then leave the factorization as: $(2u^2 - 1)(u + 6)$

Thus, the fully factored form of $2u^3 + 12u^2 - u - 6$ is:

$(2u^2 - 1)(u + 6)$

Would you like more details or have any questions?

Related Questions

1. How do you recognize terms that can be grouped for factoring?
2. What are common mistakes when factoring polynomials by grouping?
3. How can you factor a polynomial using synthetic division?
4. What is the difference of squares and how do you factor it?
5. How do you handle factoring polynomials with more than four terms?

Tip:

When factoring by grouping, always check for a common factor among all terms before grouping, as it can simplify the problem significantly.