Multiply $ \color{blue}{x} $ by $ \left( x+5\right) $ $$ \color{blue}{x} \cdot \left( x+5\right) = x^2+5x $$

Remove the parentheses by changing the sign of each term within them.

$$ - \left( x^2+5x \right) = -x^2-5x $$

Multiply $ \dfrac{1}{20} $ by $ x $ to get $ \dfrac{ x }{ 20 } $.

Step 1: Write $ x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{1}{20} \cdot x & \xlongequal{\text{Step 1}} \frac{1}{20} \cdot \frac{x}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 1 \cdot x }{ 20 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ x }{ 20 } \end{aligned} $$

Multiply $ \dfrac{19}{100} $ by $ x+5 $ to get $ \dfrac{ 19x+95 }{ 100 } $.

Step 1: Write $ x+5 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{19}{100} \cdot x+5 & \xlongequal{\text{Step 1}} \frac{19}{100} \cdot \frac{x+5}{\color{red}{1}} \xlongequal{\text{Step 2}} \frac{ 19 \cdot \left( x+5 \right) }{ 100 \cdot 1 } = \\[1ex] & \xlongequal{\text{Step 3}} \frac{ 19x+95 }{ 100 } \end{aligned} $$

Add $ \dfrac{x}{20} $ and $ \dfrac{19}{100} $ to get $ \dfrac{ \color{purple}{ 5x+19 } }{ 100 }$.

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

$$ \begin{aligned} \frac{x}{20} + \frac{19}{100} & = \frac{ x \cdot \color{blue}{ 5 }}{ 20 \cdot \color{blue}{ 5 }} + \frac{ 19 }{ 100 } = \\[1ex] &=\frac{ \color{purple}{ 5x } }{ 100 } + \frac{ \color{purple}{ 19 } }{ 100 }=\frac{ \color{purple}{ 5x+19 } }{ 100 } \end{aligned} $$

Subtract $ \dfrac{19x+95}{100} $ from $ 1-x^2-5x $ to get $ \dfrac{ \color{purple}{ -100x^2-519x+5 } }{ 100 }$.

Step 1: Write $ 1-x^2-5x $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: To subtract raitonal expressions, both fractions must have the same denominator.
We can create a common denominator by multiplying the first fraction by $\color{blue}{ 100 }$.

$$ \begin{aligned} 1-x^2-5x- \frac{19x+95}{100} & \xlongequal{\text{Step 1}} \frac{1-x^2-5x}{\color{red}{1}} - \frac{19x+95}{100} = \frac{ \left( 1-x^2-5x \right) \cdot \color{blue}{ 100 }}{ 1 \cdot \color{blue}{ 100 }} - \frac{ 19x+95 }{ 100 } = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \color{purple}{ 100-100x^2-500x } }{ 100 } - \frac{ \color{purple}{ 19x+95 } }{ 100 }=\frac{ \color{purple}{ 100-100x^2-500x - \left( 19x+95 \right) } }{ 100 } = \\[1ex] &=\frac{ \color{purple}{ -100x^2-519x+5 } }{ 100 } \end{aligned} $$

Multiply $ \dfrac{5x+19}{100} $ by $ x^3+9x^2+x+1 $ to get $ \dfrac{5x^4+64x^3+176x^2+24x+19}{100} $.

Step 1: Write $ x^3+9x^2+x+1 $ as a fraction by putting $ \color{red}{ 1 } $ in the denominator.

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{5x+19}{100} \cdot x^3+9x^2+x+1 & \xlongequal{\text{Step 1}} \frac{5x+19}{100} \cdot \frac{x^3+9x^2+x+1}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( 5x+19 \right) \cdot \left( x^3+9x^2+x+1 \right) }{ 100 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ 5x^4+45x^3+5x^2+5x+19x^3+171x^2+19x+19 }{ 100 } = \\[1ex] &= \frac{5x^4+64x^3+176x^2+24x+19}{100} \end{aligned} $$

Multiply $ \dfrac{-100x^2-519x+5}{100} $ by $ x^2+4x+1 $ to get $ \dfrac{-100x^4-919x^3-2171x^2-499x+5}{100} $.

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

Step 2: Multiply numerators and denominators.

Step 3: Simplify numerator and denominator.

$$ \begin{aligned} \frac{-100x^2-519x+5}{100} \cdot x^2+4x+1 & \xlongequal{\text{Step 1}} \frac{-100x^2-519x+5}{100} \cdot \frac{x^2+4x+1}{\color{red}{1}} = \\[1ex] & \xlongequal{\text{Step 2}} \frac{ \left( -100x^2-519x+5 \right) \cdot \left( x^2+4x+1 \right) }{ 100 \cdot 1 } \xlongequal{\text{Step 3}} \frac{ -100x^4-400x^3-100x^2-519x^3-2076x^2-519x+5x^2+20x+5 }{ 100 } = \\[1ex] &= \frac{-100x^4-919x^3-2171x^2-499x+5}{100} \end{aligned} $$

Add $ \dfrac{5x^4+64x^3+176x^2+24x+19}{100} $ and $ \dfrac{-100x^4-919x^3-2171x^2-499x+5}{100} $ to get $ \dfrac{-95x^4-855x^3-1995x^2-475x+24}{100} $.

To add expressions with the same denominators, we add the numerators and write the result over the common denominator.

$$ \begin{aligned} \frac{5x^4+64x^3+176x^2+24x+19}{100} + \frac{-100x^4-919x^3-2171x^2-499x+5}{100} & = \frac{5x^4+64x^3+176x^2+24x+19}{\color{blue}{100}} + \frac{-100x^4-919x^3-2171x^2-499x+5}{\color{blue}{100}} = \\[1ex] &=\frac{ 5x^4+64x^3+176x^2+24x+19 + \left( -100x^4-919x^3-2171x^2-499x+5 \right) }{ \color{blue}{ 100 }} = \\[1ex] &= \frac{-95x^4-855x^3-1995x^2-475x+24}{100} \end{aligned} $$