To simplify the expression
[tex]\[
\frac{5x - 1}{x + 3} + \frac{9}{x(x + 3)},
\][/tex]
we first find a common denominator for the fractions. The common denominator is [tex]\(x(x + 3)\)[/tex].
We begin by rewriting each fraction with this common denominator:
1. The first fraction [tex]\(\frac{5x - 1}{x + 3}\)[/tex]:
To convert [tex]\(\frac{5x - 1}{x + 3}\)[/tex] to have the common denominator [tex]\(x(x + 3)\)[/tex], we multiply both the numerator and the denominator by [tex]\(x\)[/tex]:
[tex]\[
\frac{5x - 1}{x + 3} = \frac{x(5x - 1)}{x(x + 3)} = \frac{5x^2 - x}{x(x + 3)}.
\][/tex]
2. The second fraction [tex]\(\frac{9}{x(x + 3)}\)[/tex] already has the common denominator:
[tex]\[
\frac{9}{x(x + 3)}.
\][/tex]
Now, we add the fractions:
[tex]\[
\frac{5x^2 - x}{x(x + 3)} + \frac{9}{x(x + 3)} = \frac{5x^2 - x + 9}{x(x + 3)}.
\][/tex]
Thus, the combined fraction is:
[tex]\[
\frac{5x^2 - x + 9}{x(x + 3)}.
\][/tex]
So, the simplified form of the given expression is:
[tex]\[
\frac{5x^2 - x + 9}{x(x + 3)}.
\][/tex]
Finally, according to your question's format, this means our combined fraction has:
Numerator: [tex]\( 5x^2 - x + 9 \)[/tex]
Denominator: [tex]\( x(x + 3) \)[/tex]
