The given equation is a cubic equation:

\[
2x^3 + 2x^2 - 52x + 48 = 0
\]

To solve this equation, let's follow these steps:

1. **Factor out the common factor**: Notice that all terms are divisible by 2.

   \[
   2(x^3 + x^2 - 26x + 24) = 0
   \]

   Simplify by dividing both sides by 2:

   \[
   x^3 + x^2 - 26x + 24 = 0
   \]

2. **Use the Rational Root Theorem** to find possible rational roots: The possible rational roots are factors of the constant term (24) divided by factors of the leading coefficient (1). These factors are:

   \[
   \pm 1, \pm 2, \pm 3, \pm 4, \pm 6, \pm 8, \pm 12, \pm 24
   \]

3. **Test these roots using synthetic division or substitution** to find one root. Let’s test these values to find a root.

Let's start with synthetic division to find the first root and factorize the cubic expression.The roots of the equation \( x^3 + x^2 - 26x + 24 = 0 \) are:

\[
x = -6, \quad x = 1, \quad x = 4
\]

Thus, the complete solution to the original equation \( 2x^3 + 2x^2 - 52x + 48 = 0 \) includes these three roots:

\[
x = -6, \quad x = 1, \quad x = 4
\]

Would you like further details or have any questions?

Here are some related questions you might explore:

1. How do you perform synthetic division to verify these roots?
2. What is the Rational Root Theorem, and how is it used in solving polynomial equations?
3. How can you factorize a cubic polynomial once a root is found?
4. What are the possible scenarios when a cubic equation has complex roots?
5. How can you graphically represent the roots of a cubic function?

**Tip:** When solving higher-degree polynomials, always start by simplifying the equation, and look for rational roots to factorize the polynomial step-by-step.

Explore how to solve the cubic equation 2x^3 + 2x^2 − 52x + 48 = 0 using synthetic division and the Rational Root Theorem. This detailed guide walks through factorization and root finding, perfect for high school students in Grades 10-12.

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