What is the base od this number system

the product of 44 and 11 gives 1034 as answer. what is the base of the system??

Think of how we write numbers. Supposing we have a base of 10, what would the number 35 be? It'd be 3 * 10^1 + 5 * 10^0 => 3 * 10 + 5 * 1 => 3 * 10 + 5
What would it be with a base of 9?

3 * 9^1 + 5 * 10^0 => 3 * 9 + 5

Using that format, we can construct an algebraic equation:

(4x + 4) * (1x + 1) = 1x^3 + 0x^2 + 3x + 4
4 * (x + 1) * (x + 1) = x^3 + 3x + 4
4 * (x^2 + 2x + 1) = x^3 + 3x + 4
4x^2 + 8x + 4 = x^3 + 3x + 4
4x^2 + 8x = x^3 + 3x
0 = x^3 - 4x^2 + 3x - 8x
0 = x^3 - 4x^2 - 5x
0 = x * (x^2 - 4x - 5)
0 = x * (x - 5) * (x + 1)

x = 0 , x = 5 , x = -1

x can't be 0, and a base of -1 doesn't make any sense, but x = 5 is our last option. Let's try it:

4 * 5 + 4 = 24
1 * 5 + 1 = 6

1034 =>
1 * 5^3 + 3 * 5 + 4 =>
125 + 15 + 4 =>
144

24 * 6 => 144

It works

(4b + 4)(b+1) = b³ + 3b + 4
4b² + 8b + 4 = b³ + 3b + 4
b³ - 4b² - 5b = 0
b(b²- 4b - 5) = 0
b(b-5)(b+1) = 0
b = -1, 0, 5

Check: base = 5
(4*5+4)(5+1) = 5³ + 3*5 + 4
24*6 = 125+15+4
144 = 144

Check: base = -1
[4*(-1) + 4]*(-1+1) = (-1)³ + 3*(-1) + 4
0*0 = -1-3+4
0 = 0

Either -1 or 5 could be the base and satisfies the conditions but number bases are conventionally positive integers so choose 5 as the base.

Base of the system = 5

Suppose the base is b.
In base 10 the numbers are 4b + 4 and 11 = b + 1 and b³ + 3b + 4
(4b + 4)(b + 1) = b³ + 3b + 4
4b² + 8b + 4 = b³ + 3b + 4
4b² + 8b = b³ + 3b
4b + 8 = b² + 3
b² - 4b - 5 = 0
(b - 5)(b + 1) = 0
b = 5 . . . ignore the negative solution
It's base 5.

hexadecimal