To solve the polynomial equation \( 3x^4 - 12x^3 - 72x^2 + 303x - 60 = 0 \), we can use various methods such as factoring, synthetic division, or applying the Rational Root Theorem. Let's start by simplifying and searching for rational roots.

1. **Divide the equation by 3** to make calculations easier:

   \[
   x^4 - 4x^3 - 24x^2 + 101x - 20 = 0
   \]

2. **Use the Rational Root Theorem** to test potential rational roots, which could be factors of -20 (the constant term). The possible rational roots are \( \pm 1, \pm 2, \pm 4, \pm 5, \pm 10, \pm 20 \).

3. **Test these roots** one by one in the polynomial. 

Let's test \( x = 1 \):

\[
1^4 - 4(1)^3 - 24(1)^2 + 101(1) - 20 = 1 - 4 - 24 + 101 - 20 = 54 \quad (\text{not a root})
\]

Now test \( x = 2 \):

\[
2^4 - 4(2)^3 - 24(2)^2 + 101(2) - 20 = 16 - 32 - 96 + 202 - 20 = 70 \quad (\text{not a root})
\]

Continue testing roots until a suitable one is found. 

After testing \( x = 5 \):

\[
5^4 - 4(5)^3 - 24(5)^2 + 101(5) - 20 = 625 - 500 - 600 + 505 - 20 = 10 \quad (\text{not a root})
\]

Upon further testing, you might find that \( x = 4 \):

\[
4^4 - 4(4)^3 - 24(4)^2 + 101(4) - 20 = 256 - 256 - 384 + 404 - 20 = 0 \quad (\text{is a root})
\]

4. **Use synthetic division** with \( x = 4 \):

After synthetic division, you get a cubic polynomial:

\[
x^3 + 0x^2 - 24x + 5 = 0
\]

5. **Repeat the process** to find the roots of the cubic polynomial using either synthetic division again or numerical methods.

Let me know if you want a step-by-step solution of the cubic polynomial or any specific method to proceed with finding the remaining roots! 

### Questions:
1. Would you like to explore a specific method for finding the roots?
2. Do you want to see the synthetic division process in detail?
3. Are you interested in numerical approximations for the roots?
4. Would you like to know about the graphical representation of the polynomial?
5. Do you have other equations to solve or compare with this one?

### Tip:
Always check for simple roots first, as they can significantly simplify your calculations.

Explore the step-by-step solution for the polynomial equation 3x^4 - 12x^3 - 72x^2 + 303x - 60 = 0. This guide covers essential algebraic concepts, including the Rational Root Theorem and synthetic division, suitable for high school students in Grades 10-12.

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