To simplify the expression

$\frac{6(x+1)^2 - 2(x-2)(x+1)}{(x+1)^3}$

let's go through it step by step.

Step 1: Expand the Terms in the Numerator

The numerator is $6(x+1)^2 - 2(x-2)(x+1)$.

1. Expand $6(x+1)^2$: $6(x+1)^2 = 6(x^2 + 2x + 1) = 6x^2 + 12x + 6$

2. Expand $2(x-2)(x+1)$: $2(x-2)(x+1) = 2(x^2 - x - 2) = 2x^2 - 2x - 4$

So, the numerator becomes: $6x^2 + 12x + 6 - (2x^2 - 2x - 4)$

Step 2: Simplify the Numerator

Combine like terms in the numerator:

$6x^2 + 12x + 6 - 2x^2 + 2x + 4 = (6x^2 - 2x^2) + (12x + 2x) + (6 + 4)$

$= 4x^2 + 14x + 10$

So the expression simplifies to:

$\frac{4x^2 + 14x + 10}{(x+1)^3}$

Step 3: Factor the Numerator if Possible

Let's factor $4x^2 + 14x + 10$:

$4x^2 + 14x + 10 = 2(2x^2 + 7x + 5)$

Factoring $2x^2 + 7x + 5$: $2x^2 + 7x + 5 = (2x + 5)(x + 1)$

Thus, the numerator becomes:

$4x^2 + 14x + 10 = 2(2x + 5)(x + 1)$

Step 4: Cancel Terms

Now, substitute back into the expression:

$\frac{2(2x + 5)(x + 1)}{(x + 1)^3}$

Cancel $(x + 1)$ from the numerator and denominator:

$= \frac{2(2x + 5)}{(x + 1)^2}$

Final Answer

The simplified form is:

$\frac{2(2x + 5)}{(x + 1)^2}$

Would you like further details on any step, or do you have questions about this simplification?

Here are five additional questions to consider:

1. How can you factor quadratic expressions effectively?
2. What other forms of simplification can help with rational expressions?
3. When is it helpful to expand expressions before simplifying?
4. How do you handle undefined points in simplified expressions?
5. What are common mistakes to avoid in polynomial simplification?

Tip: Always check if you can factor out terms before expanding—they might simplify directly!