Question

what is the simplest form of the expression below?\\(\frac{2x^2 - 10x - 28}{6x} \times \frac{6}{x - 7}\\)\
a. \\(\frac{2x + 4}{x}\\)\
b. \\(6\\)\
c. \\(\frac{12x + 24}{6x}\\)\
d. \\(\frac{2x^2 - 10x - 168}{6x^2 - 42x}\\)

Explanation:

Step1: Factor the numerator of the first fraction

First, factor out the greatest common factor from \(2x^2 - 10x - 28\). The GCF of 2, -10, and -28 is 2. So we have:
\(2x^2 - 10x - 28 = 2(x^2 - 5x - 14)\)
Now, factor the quadratic \(x^2 - 5x - 14\). We need two numbers that multiply to -14 and add to -5. Those numbers are -7 and 2. So:
\(x^2 - 5x - 14=(x - 7)(x + 2)\)
Thus, \(2x^2 - 10x - 28 = 2(x - 7)(x + 2)\)

Step2: Multiply the two fractions

The original expression is \(\frac{2x^2 - 10x - 28}{6x}\times\frac{6}{x - 7}\). Substitute the factored form of the numerator of the first fraction:
\(\frac{2(x - 7)(x + 2)}{6x}\times\frac{6}{x - 7}\)
Now, cancel out common factors. The \((x - 7)\) terms cancel, the 6 in the numerator of the second fraction and the 6 in the denominator of the first fraction cancel:
\(\frac{2\cancel{(x - 7)}(x + 2)}{\cancel{6}x}\times\frac{\cancel{6}}{\cancel{(x - 7)}}=\frac{2(x + 2)}{x}\)
Simplify the numerator: \(2(x + 2)=2x + 4\)
So the simplified form is \(\frac{2x + 4}{x}\)

Answer:

A. \(\frac{2x + 4}{x}\)