Question 1a

3 12 5^{x^{3} - 5 x^{2} + 11 x - 3} = 4 5^{4 x^{2} + 12}

Change the $125 \to 5^{3}$ :

3 5^{3 (x^{3} - 5 x^{2} + 11 x - 3)} = 4 5^{4 x^{2} + 12}

Change the to index form:

(5^{3 (x^{3} - 5 x^{2} + 11 x - 3)})^{\frac{1}{3}} = (5^{4 x^{2} + 12})^{\frac{1}{4}} ⟺ 5^{x^{3} - 5 x^{2} + 11 x - 3} = 5^{x^{2} + 3}

After simplification, and after using the index rule:

x^{3} - 5 x^{2} + 11 x - 3 = x^{2} + 3 ⟺ x^{3} - 6 x^{2} + 11 x - 6 = 0

We get a cubic equation.

Solving synthetically with 1 (guess):

111 - 6 1 - 5 11 - 5 6 - 6 60

$1$ is a solution, and hence the other solutions are $2$ and $3$ !

⟹ x = 1, 2, 3

ℕBTS