# Simplify the expression [(x^2-6x+5)/(x+3)]*[(x^2-9)/(x-5).

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Let E = (x^2-6x+5) / (x+3) * (x^2-9)/(x-5)

First we will multiply numerators and denominators:

==> E = (x^2-6x+5)*(x^2-9)/(x+3)(x-5)

Now we will factor the numerator:

==> E = (x-5)(x-1)(x-3)(x+3)/(x+3)(x-5)

Now we will reduce similar terms.

==> E = (x-1)(x-3)

**==> E = (x-1)(x-3) = x^2 -4x + 3**

To simplify {(x^2-6x+5)/(x+3)}{(x^2-9)/(x+5)}.

Solution:

x^2--6x+5 = (x-1)(x-5)

x^2-9 = (x-3)(x+3)

So the given expression is rewritten sing the above factors:

{(x-1)(x-5)/(x+3)}/{(x-3)(x+3)/(x-5)}

=(x-1)(x-5)(x-3)(x+3)/{(x+3)((x-5)

=(x-1)(x-3)

= x^2-4x+3.

Therefore [(x^2-6x+5)/(x+3)]*{(x^2-9)/(x-5)} = x^2-4x+3.

We'll factorize all fractions in distinct factors. For this reason, we'll determine the roots of the first numerator:

x^2-6x+5 = 0

We'll apply quadratic formula:

x1 = [6+sqrt(36 - 20)]/2

x1 = (6+4)/2

x1 = 5

x2 = (6-4)/2

x2 = 1

The equation will be written as:

x^2-6x+5= (x-5)(x-1)

We'll write the numerator of the 2nd factor as a product:

x^2 - 9 = (x-3)(x+3)

We'll re-write the factorised expression:

[(x-5)(x-1)/(x+3)]*[(x-3)(x+3)/(x-5)]

We'll cancel common factors:

(x-1)(x-3)

We'll leave the factors as they are and the simplified expression is:

(x-1)(x-3)

We'll remove the brackets:

(x-1)(x-3) = x^2 - x - 3x + 3

We'll combine like terms:

(x-1)(x-3) = x^2 - 4x + 3

**The simplified expression is:**

**x^2 - 4x + 3**