4x-3/x2=m could be interpreted in two ways:
(1) 4x-(3/x2)=m,
4x3-3=mx2,
4x3-mx2-3=0.
Since x=-4 is a root (a zero) then x+4 is a factor. Synthetic division:
-4 | 4 -m 0 -3
4 -16 64+4m | -256-16m
4 -16-m 64+4m | 0 ⇒-259-16m=0, m=-259/16; the other factor is the quadratic:
4x2-(16-m)x+4m+64 which becomes 4x2+3x/16-¾ after substituting for m.
CHECK: (x+4)(4x2+3x/16-¾)=4x3+3x2/16-3x/4+16x2+3x/4-3=4x3+259x2/16-3=0,
4x+259/16-3/x2=0, 4x-(3/x2)=-259/16=m. So this result is confirmed.
To find other roots we need to solve the quadratic 4x2+3x/16-¾=0.
We can use the quadratic formula:
x=(-3/16±√(9/256+12))/8=(-0.1875±3.4692)/8=0.4102 or -0.4571 (approx) for the other two roots.
(2) (4x-3)/x2=m,
4x-3=mx2,
mx2-4x+3=0. Since x=-4 is a solution, we can use synthetic division:
-4 | m -4 3
m -4m | 16m+16
m -4m-4 | 16m+19⇒16m+19=0, m=-19/16.
Another way, using x=-4 in the equation: mx2-4x+3=0⇒16m+16=-3, m=-19/16.
CHECK: (x+4)(-19x/16+¾)=-19x2/16+3x/4-19x/4+3=-19x216-4x+3=0, 4x-3=-19x2/16, (4x-3)/x2=-19/16, confirming the result.
The other root is x=12/19 (that is, mx-4(m+1), then substitute for m, as given by the quotient of the synthetic division).
NOTE: If m=1 then 4x-3=x2, x2-4x+3=0=(x-3)(x-1). This would seem an attractive solution for m, but for the fact that x=-4 is not a root.