add the fractions 4/x and (x+1)/(x-5)

The original denominators must be nonzero:

x \in ℝ \ {0, 5}

Let's sum the fractions:

\frac{4}{x} + \frac{x + 1}{x - 5}

ⓘ Add fractions by using a common denominator and then adding their numerators:

\frac{4}{x} + \frac{x + 1}{x - 5}

\frac{4 \cdot (x - 5) + (x + 1) \cdot x}{x \cdot (x - 5)}

\frac{4 \cdot (x - 5) + (x + 1) \cdot x}{x \cdot (x - 5)}

ⓘ Multiply each term of the sum by the other factor, using the distributive law:

4 \cdot (x - 5)

4 x - 20

(x + 1) \cdot x

x^{2} + x

\frac{4 x - 20 + x^{2} + x}{x \cdot (x - 5)}

ⓘ Group like terms by adding their coefficients:

4 x + x

5 x

\frac{5 x - 20 + x^{2}}{x \cdot (x - 5)}

▶Result:

\frac{x^{2} + 5 x - 20}{x \cdot (x - 5)}