Compute the largest possible number of distinct real solutions for $x$ to the equation \[x^6+ax^5+60x^4-159x^3+240x^2+bx+c=0,\] where $a$, $b$, and $c$ are real numbers.

[i]Proposed by Tristan Shin
1. **Understanding the Polynomial and Its Degree:**
   The given polynomial is of degree 6:
   \[
   P(x) = x^6 + ax^5 + 60x^4 - 159x^3 + 240x^2 + bx + c
   \]
   A polynomial of degree 6 can have at most 6 real roots.

2. **Application of Newton's Inequalities:**
   Newton's inequalities provide conditions on the coefficients of a polynomial that can restrict the number of real roots. However, in this context, we need to consider the nature of the roots (real vs. complex).

3. **Complex Roots and Their Pairs:**
   Complex roots of polynomials with real coefficients always come in conjugate pairs. Therefore, the number of complex roots must be even.

4. **Analyzing Possible Numbers of Real Roots:**
   - **6 Real Roots:** If all roots were real, we would have 6 real roots. However, Newton's inequalities or other constraints might prevent this.
   - **5 Real Roots:** This is impossible because the remaining root would have to be complex, and complex roots must come in pairs.
   - **4 Real Roots:** This is possible if the remaining 2 roots are a pair of complex conjugates.
   - **3 Real Roots:** This is possible if the remaining 3 roots are complex, but they must form 1 real root and 1 pair of complex conjugates, which is not possible.
   - **2 Real Roots:** This is possible if the remaining 4 roots are two pairs of complex conjugates.
   - **1 Real Root:** This is possible if the remaining 5 roots are complex, but they must form 2 pairs of complex conjugates and 1 real root, which is not possible.
   - **0 Real Roots:** This is possible if all 6 roots are complex, forming 3 pairs of complex conjugates.

5. **Constructing a Polynomial with 4 Real Roots:**
   To construct a polynomial with exactly 4 real roots, we can consider:
   \[
   (x - r_1)(x - r_2)(x - r_3)(x - r_4)(x^2 + px + q)
   \]
   Here, \(r_1, r_2, r_3, r_4\) are real roots, and \(x^2 + px + q\) has complex conjugate roots. This construction is feasible and satisfies the conditions.

6. **Conclusion:**
   The largest possible number of distinct real solutions for the given polynomial is 4.

The final answer is $\boxed{4}$