Question 1209745
Here's how to factor the expression:

1. **Expand the left side:**
   (a² + b² - c²)² + 4a²bc = a⁴ + b⁴ + c⁴ + 2a²b² - 2a²c² - 2b²c² + 4a²bc

2. **Rearrange the terms strategically:**  We want to try to create perfect squares. Notice the 4a²bc term.  Let's try to incorporate it with other terms:

   a⁴ + b⁴ + c⁴ + 2a²b² - 2a²c² - 2b²c² + 4a²bc = a⁴ + b⁴ + c⁴ + 2a²b² + 2a²c² + 2b²c² - 4a²c² - 4b²c² + 4a²bc

   = a⁴ + b⁴ + c⁴ + 2a²b² + 2a²c² + 2b²c² - 4a²c² - 4b²c² + 4a²bc
   = (a² + b² + c²)² - 4c²(a² + b² - ab)

   This doesn't seem to be getting us anywhere useful. Let's try a different approach.

3. **Try grouping terms differently:**
    a⁴ + b⁴ + c⁴ + 2a²b² - 2a²c² - 2b²c² + 4a²bc = a⁴ + 2a²b² + b⁴ - 2a²c² + 4a²bc - 2b²c² + c⁴
    = (a² + b²)² - 2c²(a² - 2abc/2c² + b²) + c⁴
    = (a² + b²)² - 2c²(a² + b² - 2abc/2c²) + c⁴

Let's try another approach.  We can rewrite the expression as:

(a² + b² - c²)² + 4a²bc = a⁴ + b⁴ + c⁴ + 2a²b² - 2a²c² - 2b²c² + 4a²bc

Notice if we rewrite 4a²bc as 2a²bc + 2a²bc, we can rearrange:

= a⁴ + b⁴ + c⁴ + 2a²b² - 2a²c² - 2b²c² + 2a²bc + 2a²bc
= a⁴ + b⁴ + c⁴ + 2a²b² + 2a²bc - 2a²c² + 2a²bc - 2b²c²
= a⁴ + b⁴ + c⁴ + 2a²(b² + bc - c²) + 2a²bc - 2b²c²
= (a² + b² + c² + 2ab)(a² + b² + c² - 2ac)(a² + b² + c² - 2bc)

This also doesn't immediately lead to the desired factorization.

Let's look at the original expression again: (a² + b² - c²)² + 4a²bc

If we let c = 0, we get (a² + b²)² which factors into (a + ib)(a - ib)(a + ib)(a - ib).  This doesn't seem to help much either.

After trying several approaches, the given expression doesn't seem to readily factor into the form (a + ...)(a + ...)(a + ...)(a + ...). It's possible there's a typo in the original problem, or it might require a more complex factorization involving complex numbers.

Given the difficulty and the lack of a clear path to the requested factorization with real numbers, it's possible that the intended factorization involves complex numbers or that the problem itself has a typo.  Without further information or context, it's difficult to proceed.