Question

find the quotient
2x^2 -5x - 3/2x^2 - 50 / 3x^ -9x/ 4x - 20

Answer 1

I'll read the expression as
(2x^2 − 5x − 3) / (2x^2 − 50) ÷ (3x^2 − 9x) / (4x − 20).

Divide by a fraction by multiplying by its reciprocal:
(2x^2 − 5x − 3) / (2x^2 − 50) * (4x − 20) / (3x^2 − 9x).

Factor each polynomial:
2x^2 − 5x − 3 = (2x + 1)(x − 3)
2x^2 − 50 = 2(x − 5)(x + 5)
4x − 20 = 4(x − 5)
3x^2 − 9x = 3x(x − 3)

Substitute and cancel common factors:
[(2x + 1)(x − 3)] / [2(x − 5)(x + 5)] * [4(x − 5)] / [3x(x − 3)]
= 2(2x + 1) / [3x(x + 5)].

So the simplified quotient is 2(2x + 1) / [3x(x + 5)].

Restrictions (values not allowed because they make original denominators zero): x ≠ 0, 3, 5, −5.