Sure, let's work through the simplification step-by-step.
We start with the following expression:
[tex]\[ \frac{5x - 1}{x + 3} + \frac{9}{x(x + 3)} \][/tex]
First, let's find a common denominator. The common denominator for the two fractions is [tex]\( x(x + 3) \)[/tex].
Next, we rewrite each fraction with the common denominator:
[tex]\[ \frac{5x - 1}{x + 3} = \frac{(5x - 1) \cdot x}{(x + 3) \cdot x} = \frac{5x^2 - x}{x(x + 3)} \][/tex]
[tex]\[ \frac{9}{x(x + 3)} \][/tex]
Now we can add the two fractions with the common denominator [tex]\( x(x + 3) \)[/tex]:
[tex]\[ \frac{5x^2 - x}{x(x + 3)} + \frac{9}{x(x + 3)} = \frac{5x^2 - x + 9}{x(x + 3)} \][/tex]
The expression in the numerator is now [tex]\( 5x^2 - x + 9 \)[/tex].
So the simplified expression is:
[tex]\[ \frac{5x^2 - x + 9}{x(x + 3)} \][/tex]
Therefore, the simplified form of the given expression is:
[tex]\[ \frac{5x^2 - x + 9}{x(x + 3)} \][/tex]
