Multiply each term of $ \left( \color{blue}{21x^2+10x+8}\right) $ by each term in $ \left( 21x^2+10x+8\right) $.

$$ \left( \color{blue}{21x^2+10x+8}\right) \cdot \left( 21x^2+10x+8\right) = \\ = 441x^4+210x^3+168x^2+210x^3+100x^2+80x+168x^2+80x+64 $$

Combine like terms:

$$ 441x^4+ \color{blue}{210x^3} + \color{red}{168x^2} + \color{blue}{210x^3} + \color{green}{100x^2} + \color{orange}{80x} + \color{green}{168x^2} + \color{orange}{80x} +64 = \\ = 441x^4+ \color{blue}{420x^3} + \color{green}{436x^2} + \color{orange}{160x} +64 $$

Multiply each term of $ \left( \color{blue}{441x^4+420x^3+436x^2+160x+64}\right) $ by each term in $ \left( 21x^2+10x+8\right) $.

$$ \left( \color{blue}{441x^4+420x^3+436x^2+160x+64}\right) \cdot \left( 21x^2+10x+8\right) = \\ = 9261x^6+4410x^5+3528x^4+8820x^5+4200x^4+3360x^3+9156x^4+4360x^3+3488x^2+3360x^3+1600x^2+1280x+1344x^2+640x+512 $$

Combine like terms:

$$ 9261x^6+ \color{blue}{4410x^5} + \color{red}{3528x^4} + \color{blue}{8820x^5} + \color{green}{4200x^4} + \color{orange}{3360x^3} + \color{green}{9156x^4} + \color{blue}{4360x^3} + \color{red}{3488x^2} + \color{blue}{3360x^3} + \color{green}{1600x^2} + \color{orange}{1280x} + \color{green}{1344x^2} + \color{orange}{640x} +512 = \\ = 9261x^6+ \color{blue}{13230x^5} + \color{green}{16884x^4} + \color{blue}{11080x^3} + \color{green}{6432x^2} + \color{orange}{1920x} +512 $$

Subtract $ \dfrac{84x+60}{21x^2+10x+8} $ from $ \dfrac{42x+10}{9261x^6+13230x^5+16884x^4+11080x^3+6432x^2+1920x+512} $ to get $ \dfrac{ \color{purple}{ -37044x^5-61740x^4-61824x^3-39600x^2-14934x-3830 } }{ 9261x^6+13230x^5+16884x^4+11080x^3+6432x^2+1920x+512 }$.

To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the second fraction by $\color{blue}{441x^4+420x^3+436x^2+160x+64}$.

$$ \begin{aligned} \frac{42x+10}{9261x^6+13230x^5+16884x^4+11080x^3+6432x^2+1920x+512} - \frac{84x+60}{21x^2+10x+8} & = \frac{ 42x+10 }{ 9261x^6+13230x^5+16884x^4+11080x^3+6432x^2+1920x+512 } - \frac{ \left( 84x+60 \right) \cdot \color{blue}{ \left( 441x^4+420x^3+436x^2+160x+64 \right) }}{ \left( 21x^2+10x+8 \right) \cdot \color{blue}{ \left( 441x^4+420x^3+436x^2+160x+64 \right) }} = \\[1ex] &=\frac{ \color{purple}{ 42x+10 } }{ 9261x^6+13230x^5+16884x^4+11080x^3+6432x^2+1920x+512 } - \frac{ \color{purple}{ 37044x^5+35280x^4+36624x^3+13440x^2+5376x+26460x^4+25200x^3+26160x^2+9600x+3840 } }{ 9261x^6+13230x^5+16884x^4+11080x^3+6432x^2+1920x+512 } = \\[1ex] &=\frac{ \color{purple}{ 42x+10 - \left( 37044x^5+35280x^4+36624x^3+13440x^2+5376x+26460x^4+25200x^3+26160x^2+9600x+3840 \right) } }{ 9261x^6+13230x^5+16884x^4+11080x^3+6432x^2+1920x+512 } = \\[1ex] &=\frac{ \color{purple}{ -37044x^5-61740x^4-61824x^3-39600x^2-14934x-3830 } }{ 9261x^6+13230x^5+16884x^4+11080x^3+6432x^2+1920x+512 } \end{aligned} $$