2x^3-x^2-6x / 2x^2-7x+6
To simplify the given expression, we can start by factoring both the numerator and the denominator.
Let's first factor the numerator:
2x^3 - x^2 - 6x
We can factor out the common factor of x:
x(2x^2 - x - 6)
Now, let's factor the quadratic term inside the parentheses:
x(2x^2 - 4x + 3x - 6)
Next, we can factor by grouping:
x((2x^2 - 4x) + (3x - 6))
Now, we factor out the common factors from each group:
x(2x(x - 2) + 3(x - 2))
Notice that we have a common binomial factor of (x - 2):
x(x - 2)(2x + 3)
Now, let's factor the denominator:
2x^2 - 7x + 6
We can find two numbers whose product is 6 and whose sum is -7. These numbers are -1 and -6.
Now, we can rewrite the denominator as:
2x^2 - x - 6x + 6
Next, we group the terms and factor by grouping:
(x(2x - 1) - 2(3x - 1))
Again, we have a common binomial factor of (2x - 1):
(2x - 1)(x - 2)
Now, we can simplify the expression by canceling out any common factors in the numerator and denominator:
x(x - 2)(2x + 3) / (2x - 1)(x - 2)
Finally, we can further simplify the expression by canceling out the common binomial factor of (x - 2):
x(2x + 3) / (2x - 1)