The given expression to factor by grouping is:

$x^3 + 4x^2 - 9x - 36$

We will attempt to group terms and factor the expression.

Step 1: Group the terms

Group the terms in pairs:

$(x^3 + 4x^2) - (9x + 36)$

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

For the first group $x^3 + 4x^2$, the GCF is $x^2$:

$x^2(x + 4)$

For the second group $-9x - 36$, the GCF is $-9$:

$-9(x + 4)$

Step 3: Factor the common binomial

Now the expression looks like this:

$x^2(x + 4) - 9(x + 4)$

Since both terms contain $(x + 4)$, we can factor that out:

$(x + 4)(x^2 - 9)$

Step 4: Factor the difference of squares

The expression $x^2 - 9$ is a difference of squares, which can be factored as:

$x^2 - 9 = (x - 3)(x + 3)$

Final factored form:

$(x + 4)(x - 3)(x + 3)$

The factored form of the given expression is:

$(x + 4)(x - 3)(x + 3)$

Would you like any further details or have any questions?

Here are some related questions:

1. What is the difference of squares rule?
2. How do you identify common factors in algebraic expressions?
3. Can you factor a cubic equation directly without grouping?
4. How does factoring help in solving polynomial equations?
5. What are other methods for factoring polynomials?

Tip: Always check for common factors before applying advanced factoring methods, as this simplifies the expression early on.