4i^6+3i/1-2i-(9+4i)

$$\small{\text{ $ \boxed{ \dfrac{4i^6+3i } {1-2i} -(9+4i) =\ ? } $ }}\\\\ \small{\text{ $ i^2 = -1 \qquad i^4 = i^2 \cdot i^2 = (-1)\cdot (-1) = 1 \qquad i^6 = i^2 \cdot i^2 \cdot i^2 = (-1)\cdot (-1) \cdot (-1) = -1 $ }}\\$$

we set

$$\small{\text{$i^6 = (-1)^3 = -1$}}$$

so we have:

$$\small{\text{ $ \dfrac{4i^6+3i } {1-2i} -(9+4i) \quad | \quad \boxed{i^6=-1} $ }}\\\\ \small{\text{ $ =\dfrac{-4+3i } {1-2i} -(9+4i) $ }}\\\\ \small{\text{ $ =\dfrac{ (-4+3i) \codt(1+2i) } { (1-2i)\codt(1+2i) } -(9+4i) \quad | \quad \boxed{(a-ib)\cdot (a+ib) = a^2+b^2} $ }}\\\\ \small{\text{ $ =\dfrac{ (-4+3i) \codt(1+2i) } { 1^2+2^2 } -(9+4i) $ }}\\\\ \small{\text{ $ =\dfrac{ (-4+3i) \codt(1+2i) } { 5 } -(9+4i) $ }}\\\\ \small{\text{ $ =\dfrac{ (-4+3i) \codt(1+2i) } { 5 } -5\cdot\dfrac{ (9+4i) }{5} $ }}\\\\ \small{\text{ $ =\dfrac{ (-4+3i) \codt(1+2i)-5\codt(9+4i) } { 5 } $ }}\\\\ \small{\text{ $ =\dfrac{ -4-8i+3i+6i^2 -45 -20i } { 5 } \quad | \quad \boxed{i^2=-1} $ }}\\\\ \small{\text{ $ =\dfrac{ -4 -8i + 3i +6\cdot (-1) -45 -20i } { 5 } $ }}\\\\ \small{\text{ $ =\dfrac{ -4 -8i + 3i -6 -45 -20i } { 5 } \quad $ we put together }}\\\\ \small{\text{ $ =\dfrac{ -55 -25i } { 5 } $ }}\\\\ \small{\text{ $ =-11 -5i $ }}$$

Remember that i²= -1

i⁶=(i²)³ ; do this first ,combine like terms and then multply by the conjugate of the denominator, (1+2i).

Should be simple after that, keep combining like terms. I got -11-3i.