What is the solution to the following equation? [(x^3) + (3x^(2)) - 25x - 75 = 0] 

We are given the cubic equation:

\[
x^3 + 3x^2 - 25x - 75 = 0
\]

### Step 1: Factor the cubic equation

We will use the Rational Root Theorem to test possible rational roots. According to the theorem, the potential rational roots are factors of the constant term (\(-75\)) divided by factors of the leading coefficient (\(1\)).

Factors of \(-75\) are: \(\pm 1, \pm 3, \pm 5, \pm 15, \pm 25, \pm 75\).

Let's test \(x = -5\) by substituting it into the equation:

\[
(-5)^3 + 3(-5)^2 - 25(-5) - 75 = -125 + 75 + 125 - 75 = 0
\]

So, \(x = -5\) is a root.

### Step 2: Polynomial division

Now that we know \(x = -5\) is a root, we can divide the cubic polynomial by \(x + 5\) to reduce it to a quadratic. We perform synthetic or long division:

\[
x^3 + 3x^2 - 25x - 75 \div (x + 5)
\]

The result is:

\[
x^2 - 2x - 15
\]

### Step 3: Solve the quadratic

Now we solve the quadratic equation \(x^2 - 2x - 15 = 0\) using the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-15)}}{2(1)}
\]

\[
x = \frac{2 \pm \sqrt{4 + 60}}{2}
\]

\[
x = \frac{2 \pm \sqrt{64}}{2}
\]

\[
x = \frac{2 \pm 8}{2}
\]

So, the two solutions are:

\[
x = \frac{2 + 8}{2} = 5 \quad \text{or} \quad x = \frac{2 - 8}{2} = -3
\]

### Step 4: Final solution

The solutions to the equation are:

\[
x = -5, \quad x = 5, \quad x = -3
\]

### Summary:
The complete set of solutions is:

\[
x = -5, \quad x = 5, \quad x = -3
\]

Would you like more details on any step or have any questions?

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Here are 5 related questions to further expand on this topic:

1. How do you perform synthetic division?
2. What is the Rational Root Theorem and how is it applied?
3. How can we use graphing techniques to solve cubic equations?
4. What other methods can solve cubic equations?
5. How do we factorize a quadratic equation?

**Tip:** When solving cubic equations, always look for rational roots first using the Rational Root Theorem to simplify the equation!

What is the solution to the following equation? [(x^3) + (3x^(2)) - 25x - 75 = 0]

Discover how to solve the cubic equation x^3 + 3x^2 - 25x - 75 = 0 step by step. Learn key algebraic methods, including the Rational Root Theorem, polynomial division, and solving the resulting quadratic equation. This guide is ideal for high school students in Grades 9-12.

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