Add $-243x^5+405x^3-270x$ and $ \dfrac{90}{x} $ to get $ \dfrac{ \color{purple}{ -243x^6+405x^4-270x^2+90 } }{ x }$.

Step 1: Write $ -243x^5+405x^3-270x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ x }$.

$$ \begin{aligned} -243x^5+405x^3-270x+ \frac{90}{x} & \xlongequal{\text{Step 1}} \frac{-243x^5+405x^3-270x}{\color{red}{1}} + \frac{90}{x} = \frac{ \left( -243x^5+405x^3-270x \right) \cdot \color{blue}{ x }}{ 1 \cdot \color{blue}{ x }} + \frac{ 90 }{ x } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ -243x^6+405x^4-270x^2 } }{ x } + \frac{ \color{purple}{ 90 } }{ x }=\frac{ \color{purple}{ -243x^6+405x^4-270x^2+90 } }{ x } \end{aligned} $$

Subtract $ \dfrac{15}{x^3} $ from $ \dfrac{-243x^6+405x^4-270x^2+90}{x} $ to get $ \dfrac{ \color{purple}{ -243x^9+405x^7-270x^5+90x^3-15x } }{ x^4 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ x^3 }$ and the second by $\color{blue}{ x }$.

$$ \begin{aligned} \frac{-243x^6+405x^4-270x^2+90}{x} - \frac{15}{x^3} & = \frac{ \left( -243x^6+405x^4-270x^2+90 \right) \cdot \color{blue}{ x^3 }}{ x \cdot \color{blue}{ x^3 }} - \frac{ 15 \cdot \color{blue}{ x }}{ x^3 \cdot \color{blue}{ x }} = \\[1ex] &=\frac{ \color{purple}{ -243x^9+405x^7-270x^5+90x^3 } }{ x^4 } - \frac{ \color{purple}{ 15x } }{ x^4 } = \\[1ex] &=\frac{ \color{purple}{ -243x^9+405x^7-270x^5+90x^3-15x } }{ x^4 } \end{aligned} $$

Add $ \dfrac{-243x^9+405x^7-270x^5+90x^3-15x}{x^4} $ and $ \dfrac{1}{x^5} $ to get $ \dfrac{ \color{purple}{ -243x^{14}+405x^{12}-270x^{10}+90x^8-15x^6+x^4 } }{ x^9 }$.

To add raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $ \color{blue}{ x^5 }$ and the second by $\color{blue}{ x^4 }$.

$$ \begin{aligned} \frac{-243x^9+405x^7-270x^5+90x^3-15x}{x^4} + \frac{1}{x^5} & = \frac{ \left( -243x^9+405x^7-270x^5+90x^3-15x \right) \cdot \color{blue}{ x^5 }}{ x^4 \cdot \color{blue}{ x^5 }} + \frac{ 1 \cdot \color{blue}{ x^4 }}{ x^5 \cdot \color{blue}{ x^4 }} = \\[1ex] &=\frac{ \color{purple}{ -243x^{14}+405x^{12}-270x^{10}+90x^8-15x^6 } }{ x^9 } + \frac{ \color{purple}{ x^4 } }{ x^9 } = \\[1ex] &=\frac{ \color{purple}{ -243x^{14}+405x^{12}-270x^{10}+90x^8-15x^6+x^4 } }{ x^9 } \end{aligned} $$

Multiply $ \dfrac{-243x^{14}+405x^{12}-270x^{10}+90x^8-15x^6+x^4}{x^9} $ by $ x^4+8x^3+24x^2+32x+16 $ to get $ \dfrac{-243x^{18}-1944x^{17}-5427x^{16}-4536x^{15}+5562x^{14}+10800x^{13}+90x^{12}-7920x^{11}-2175x^{10}+2760x^9+1081x^8-472x^7-216x^6+32x^5+16x^4}{x^9} $.

Step 1: Write $ x^4+8x^3+24x^2+32x+16 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{-243x^{14}+405x^{12}-270x^{10}+90x^8-15x^6+x^4}{x^9} \cdot x^4+8x^3+24x^2+32x+16 & \xlongequal{\text{Step 1}} \frac{-243x^{14}+405x^{12}-270x^{10}+90x^8-15x^6+x^4}{x^9} \cdot \frac{x^4+8x^3+24x^2+32x+16}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( -243x^{14}+405x^{12}-270x^{10}+90x^8-15x^6+x^4 \right) \cdot \left( x^4+8x^3+24x^2+32x+16 \right) }{ x^9 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ -243x^{18}-1944x^{17}-5832x^{16}-7776x^{15}-3888x^{14}+405x^{16}+3240x^{15}+9720x^{14}+12960x^{13}+ \cancel{6480x^{12}}-270x^{14}-2160x^{13} -\cancel{6480x^{12}}-8640x^{11}-4320x^{10}+90x^{12}+720x^{11}+2160x^{10}+2880x^9+1440x^8-15x^{10}-120x^9-360x^8-480x^7-240x^6+x^8+8x^7+24x^6+32x^5+16x^4 }{ x^9 } = \\[1ex] &= \frac{-243x^{18}-1944x^{17}-5427x^{16}-4536x^{15}+5562x^{14}+10800x^{13}+90x^{12}-7920x^{11}-2175x^{10}+2760x^9+1081x^8-472x^7-216x^6+32x^5+16x^4}{x^9} \end{aligned} $$