Answer:
a. [tex]\frac{2x + 3}{x+5}[/tex]
b. x ≠ -5 (Vertical asymptote) and x ≠ 5 (Hole)
Step-by-step explanation:
Factor the numerator (Grouping):
[tex]2x^2 - 7x - 15[/tex]
Two numbers that multiply to -30 and add to -7 = -3 and 10
[tex][2x^2 - 10x] + [3x - 15][/tex]
[tex](2x + 3)(x - 5)[/tex]
Factor the denominator (Difference of Two Squares):
[tex]x^2 - 25[/tex] = [tex](x +5)(x-5)[/tex]
Factored Expression:
(x - 5) can be factored out of top and bottom as a hole-
[tex]\frac{(2x + 3)(x - 5)}{(x + 5)(x - 5)} = \frac{2x + 3}{x+5}[/tex]
Variable Restrictions:
Denominator ≠ 0
[tex]x + 5 = 0\\x = -5[/tex]
Vertical asymptote at x = -5 ⇒ x ≠ -5
