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Explanation:

Let the base be b.

∴ (4b + 4)(b + 1) = b3 + 3b + 4

∴ b3 − 4b2 − 5b = 0

Solving we get, b = 0, –1, 5

∵ Base cannot be zero or negative, so base is 5.

∴ (3111)5 = 3 × 125 + 25 + 5 + 1 = 406

Hence, option (a).

Alternatively,

Let the required base be x.

We know that the answer of 44 × 11 in the required base is 1034.

∴ As 4 occurs in the product, we can say that the base is greater than 4.

Also, 44 × 11 = 484 in base 10.

As 1034 > 484, the base is lesser than 10.

So we can represent the multiplication as follows:

44 × 11 = 44(x + 1) = 44x + 44 = 1034

4 + 4 → 3 or 13 or 23 …

As the base is less than 10, 4 + 4, which is 8 in base 10 cannot be expressed as 3 in the required base.

∴ 4 + 4 → 13 or 23…

4 + 4 → 13

∴ 8 → 13

∴ 8 → x + 3

∴ x = 5

If 4 + 4 → 23

∴ 8 → 23

∴ 8 → x + 13

∴ x = –5, which is not possible.

∴ (3111)5 = 1 × 50 + 1 × 51 + 1 × 52 + 3 × 53  = (406)10

Hence, option (a).

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