Find the value of (222224 × 444445 × 222221 + 666668) / (2222222)

When an expression is built from several numbers that are all close to one common large value, substitute a single variable x for that common value, rewrite every term as x plus or minus a small constant, and simplify the whole expression algebraically before putting the large number back in. This turns unwieldy multiplication of huge numbers into a short polynomial simplification.

Let x = 222222, so each term in the expression can be rewritten relative to x:

• 222224 = x + 2

• 444445 = 2x + 1

• 222221 = x - 1

• 666668 = 3x + 2

• 222222 squared = x squared

1. Expand (x + 2)(2x + 1) = 2x2 + x + 4x + 2 = 2x2 + 5x + 2.

2. Multiply by (x − 1): (2x2 + 5x + 2)(x − 1) = 2x3 − 2x2 + 5x2 − 5x + 2x − 2 = 2x3 + 3x2 − 3x − 2.

3. Add the remaining term (3x + 2): 2x3 + 3x2 − 3x − 2 + 3x + 2 = 2x3 + 3x2.

4. Divide the whole numerator by the denominator x2: (2x3 + 3x2) / x2 = 2x + 3.

5. Substitute x = 222222 back in: 2(222222) + 3 = 444444 + 3 = 444447.

Cross-check: test the identity (x+2)(2x+1)(x−1) + (3x+2) = 2x3 + 3x2 with a small value, say x = 2. Left side: (4)(5)(1) + 8 = 28. Right side: 2(8) + 3(4) = 16 + 12 = 28. The two sides match, and dividing 28 by x2 = 4 gives 7, which equals 2x + 3 = 2(2) + 3 = 7 — confirming the simplification is correct independent of the large substituted value.

So the expression simplifies to 2x + 3, and with x = 222222 the value is 444447.