Find the equation by factoring: 3x^3+4x^2-27x-36=0

We are asked to factor the cubic equation:

\[
3x^3 + 4x^2 - 27x - 36 = 0
\]

### Step 1: Group terms
We can attempt factoring by grouping the terms:

\[
(3x^3 + 4x^2) - (27x + 36) = 0
\]

Now, factor out the greatest common factor (GCF) from each group:

\[
x^2(3x + 4) - 9(3x + 4) = 0
\]

### Step 2: Factor by common binomial
We see that \((3x + 4)\) is a common factor:

\[
(x^2 - 9)(3x + 4) = 0
\]

### Step 3: Factor \(x^2 - 9\)
The expression \(x^2 - 9\) is a difference of squares:

\[
x^2 - 9 = (x - 3)(x + 3)
\]

Thus, the equation becomes:

\[
(x - 3)(x + 3)(3x + 4) = 0
\]

### Step 4: Solve for \(x\)
Now, set each factor equal to zero and solve:

1. \(x - 3 = 0 \Rightarrow x = 3\)
2. \(x + 3 = 0 \Rightarrow x = -3\)
3. \(3x + 4 = 0 \Rightarrow 3x = -4 \Rightarrow x = -\frac{4}{3}\)

### Final Answer:
The solutions to the equation are:

\[
x = 3, \, x = -3, \, x = -\frac{4}{3}
\]

Would you like further details on the factoring steps, or do you have any questions?

### Related Questions:
1. How do you factor higher-degree polynomials?
2. What are the rules for factoring a difference of squares?
3. How do you solve cubic equations using synthetic division?
4. Can all cubic equations be factored by grouping?
5. What methods are available when factoring by grouping doesn’t work?

**Tip:** Always check if you can factor by grouping before using more advanced methods like synthetic division or the rational root theorem.

Learn how to factor the cubic equation 3x^3 + 4x^2 - 27x - 36 using grouping and the difference of squares method. This solution covers essential algebraic concepts like factoring by grouping and the zero product property. Suitable for high school students in Grades 10-12.

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