# simplify the following: {x^2 +8x +7}/{x^2 +6x +9} *{ x^2-13x-18}/{x^2 -5x -6}

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We'll determine the roots of the numerators and denominators of both fractions:

x^2 +8x +7 = 0

We notice that the sum of the roots is -8 and the product is 7, then the roots are: x1 = -7 and x2 = -1

We can write the quadratic as a product of linear factors:

x^2 +8x +7 = (x+1)(x+7)

In the case ofÂ denominator of the 1st fraction, we'll recognize the perfect square:

x^2 +6x +9 = (x+3)^2

We'll find the roots of the numerator of the 2nd fraction:

x^2 - 13x - 18 = 0

x1 = [13+sqrt(169+72)]/2

x1 = [13+sqrt(241)]/2

x2 = [13-sqrt(241)]/2

x^2 - 13x - 18 = (x - 13/2 - sqrt241/2)(x - 13/2 + sqrt241/2)

We'll determine the roots of the 2nd denominator:

x^2 - 5x - 6 = 0

x1 = -1, x2 = 6

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

The product of fractions will be:

(x+1)*(x+7)*(x - 13/2 - sqrt241/2)*(x - 13/2 + sqrt241/2)/(x+3)^2*(x-6)*(x+1)

**We'll simplify and we'll get: (x+7)*(x^2 - 13x - 18)/(x+3)^2*(x-6)**