Question

$$\frac{4x + 1}{x - 3} - \frac{3x}{x + 5}$$

Explanation:

Step1: Find common denominator

The common denominator is $(x-3)(x+5)$.
Rewrite each fraction:
$\frac{(4x+1)(x+5)}{(x-3)(x+5)} - \frac{3x(x-3)}{(x-3)(x+5)}$

Step2: Expand numerators

Expand each product in numerator:
$(4x+1)(x+5) = 4x^2 +20x +x +5 = 4x^2 +21x +5$
$3x(x-3) = 3x^2 -9x$

Step3: Subtract numerators

Subtract the second expanded numerator from the first:
$(4x^2 +21x +5) - (3x^2 -9x) = 4x^2 +21x +5 -3x^2 +9x$

Step4: Simplify numerator

Combine like terms:
$4x^2 -3x^2 +21x +9x +5 = x^2 +30x +5$

Answer:

$\frac{x^2 + 30x + 5}{(x-3)(x+5)}$ (or $\frac{x^2 + 30x + 5}{x^2 + 2x -15}$ if expanding the denominator)