Subtract $ \dfrac{1}{x^2+25} $ from $ \dfrac{1}{x^2+9x+20} $ to get $ \dfrac{ \color{purple}{ -9x+5 } }{ x^4+9x^3+45x^2+225x+500 }$.

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^2+25 }$ and the second by $\color{blue}{ x^2+9x+20 }$.

$$ \begin{aligned} \frac{1}{x^2+9x+20} - \frac{1}{x^2+25} & = \frac{ 1 \cdot \color{blue}{ \left( x^2+25 \right) }}{ \left( x^2+9x+20 \right) \cdot \color{blue}{ \left( x^2+25 \right) }} - \frac{ 1 \cdot \color{blue}{ \left( x^2+9x+20 \right) }}{ \left( x^2+25 \right) \cdot \color{blue}{ \left( x^2+9x+20 \right) }} = \\[1ex] &=\frac{ \color{purple}{ x^2+25 } }{ x^4+25x^2+9x^3+225x+20x^2+500 } - \frac{ \color{purple}{ x^2+9x+20 } }{ x^4+25x^2+9x^3+225x+20x^2+500 } = \\[1ex] &=\frac{ \color{purple}{ x^2+25 - \left( x^2+9x+20 \right) } }{ x^4+9x^3+45x^2+225x+500 }=\frac{ \color{purple}{ -9x+5 } }{ x^4+9x^3+45x^2+225x+500 } \end{aligned} $$