A Zero Prejudice Is a Requirement

Any of the three sides may be the hypotenuse, so there are three possible equations for $x$ , namely:

$(4x-3)^2=(12-x)^2+(3x+4)^2$ with solutions $x=\dfrac{12\pm5\sqrt{42}}6$ which come to $7.401$ and $-3.401$

$(12-x)^2=(3x+4)^2+(4x-3)^2$ with solutions $x=\dfrac{-6\pm5\sqrt{30}}{12}$ which come to $1.782$ and $-2.782$

$(3x+4)^2=(4x-3)^2+(12-x)^2$ with solutions $x=\dfrac{18\pm5\sqrt2}4$ which come to $6.268$ and $2.732$

The two negative solutions give negative side lengths when substituted into $4x-3$ and are therefore not applicable.

The remaining four solutions all produce usable triangle sides, since the sum of the two smallest numbers in the triple is larger than the largest number.

$1.782$ produces sides $4.128, 10.218, 9.346$

$7.401$ produces sides $26.604, 4.599, 26.203$

$6.268$ produces sides $22.072, 5.732, 22.804$

$2.732$ produces sides $7.928, 9.268, 12.196$

So there are $\boxed4$ solutions.