Simplify the expression: (2x + 5x + 2) / (x + 1)

Combine the like terms in the numerator:

The expression becomes:

The degree of the numerator ($7x+2$) is 1.
The degree of the denominator ($x+1$) is 1.
Since the degree of the numerator is equal to the degree of the denominator, this is an improper rational function. To perform partial fraction decomposition on an improper fraction, we first perform polynomial long division.

Divide $7x+2$ by $x+1$:

This means that $\frac{7x+2}{x+1}$ can be written as $7 + \frac{-5}{x+1}$.

The expression is now in the form of a polynomial plus a proper rational function. The proper rational function $\frac{-5}{x+1}$ is already in its simplest partial fraction form.

Therefore, the partial fraction decomposition is: