Question

simplify the complex fraction.
\frac{3x^2 - 14x - 5}{\left( \frac{2x^2 - 11x + 5}{4x^2 + 8x - 5} \
ight)}

Explanation:

Step1: Rewrite as multiplication by reciprocal

$\frac{3x^2 - 14x - 5}{1} \times \frac{4x^2 + 8x - 5}{2x^2 - 11x + 5}$

Step2: Factor all quadratic polynomials

• $3x^2 -14x -5 = (3x+1)(x-5)$
• $4x^2 +8x -5 = (2x-1)(2x+5)$
• $2x^2 -11x +5 = (2x-1)(x-5)$

Substitute factored forms:
$\frac{(3x+1)(x-5)}{1} \times \frac{(2x-1)(2x+5)}{(2x-1)(x-5)}$

Step3: Cancel common factors

Cancel $(x-5)$ and $(2x-1)$ from numerator/denominator:
$(3x+1)(2x+5)$

Step4: Expand the product

$3x(2x+5) + 1(2x+5) = 6x^2 +15x +2x +5$

Step5: Combine like terms

$6x^2 + (15x+2x) +5$

Answer:

$6x^2 +17x +5$