Simplify a Rational Expression

Factor the denominator
x² - 7x + 10.

Find a pair of numbers
whose product is the constant term +10
and whose sum is the coefficient of the middle term -7.

-2⋅(-5) = +10
-2 - 5 = -7

So x² - 7x + 10
= (x - 2)(x - 5).

Factor a Quadratic Trinomial

x² - 7x + 10
= (x - 2)(x - 5)

So (given) = (x - 5) / [(x - 2)(x - 5)].

Cancel the common factor (x - 5).

Then 1/(x - 2).

So
1/(x - 2)
is the answer.

Factor the numberator
2x² + x - 1.

Find a pair of numbers
whose product is the last term, -1,
and whose sum satisfies
[number 1] + [number 2]⋅[first term's coefficient, 2] = +1.

-1⋅1 = -1
-1 + 1⋅2 = -1 + 2 = +1

So 2x² + x - 1
= (2x - 1)(x + 1).

Factor a Quadratic Trinomial: Example 5

Factor the denominator
x² - 2x - 3.

Find a pair of numbers
whose product is the constant term -3
and whose sum is the coefficient of the middle term -2.

-3⋅1 = -3
-3 + 1 = -2

So x² - 2x - 3
= (x - 3)(x + 1).

2x² + x - 1
= (2x - 1)(x + 1)

x² - 2x - 3
= (x - 3)(x + 1)

So (given) = [(2x - 1)(x + 1)] / [(x - 3)(x + 1)].

Cancel the common factor (x + 1).

Then (2x - 1)/(x - 3).

So
(2x - 1)/(x - 3)
is the answer.