# Simplify the expression. `(5x^2 + 10x - 75)/(4x^2 -24x - 28) * (2x^2 - 10x - 28)/(x^2+7x + 10)`

ajju | College Teacher | (Level 1) Adjunct Educator

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((5x^2+10x-75) (2x^2-10x-28))/((4x^2-24x-28) (x^2+7x+10))

The factors of 10 that sum to 7 are 5 and 2. So, x^2+7x+10 =(x+5)(x+2):

((5x^2+10x-75) (2x^2-10x-28))/((x+5)(x+2) (4x^2-24x-28))

Factor 4 out of 4x^2-24x-28:

((5x^2+10x-75) (2x^2-10x-28))/(4(x^2-6x-7)(x+5)(x+2))

The factors of -7 that sum to -6 are 1 and -7. So,x^2-6x-7 = (x+1)(x-7):

((5x^2+10x-75)(2x^2-10x-28))/(4(x+1)(x-7)(x+5)(x+2))

Factor 2 out of 2x^2-10x-28:

(2(x^2-5x-14)(5x^2+10x-75))/(4(x+1)(x-7)(x+5)(x+2))

The factors of -14 that sum to -5 are 2 and -7.

So,x^2-5x-14 = (x+2)(x-7):

(2(x+2)(x-7)(5x^2+10x-75))/(4(x+1)(x-7)(x+5)(x+2))

Factor 5 out of 5x^2+10x-75:

(2×5(x^2+2x-15)(x+2)(x-7))/(4(x+1)(x-7)(x+5)(x+2))

The factors of -15 that sum to 2 are 5 and -3. So, x^2+2x-15 = (x+5)(x-3):

(5×2(x+5)(x-3)(x+2)(x-7))/(4(x+1)(x-7)(x+5)(x+2))

= (5(x+5)(x-3)×2(x+2)(x-7))/(4(x+1)(x-7)(x+5)(x+2))

= ((x-7)(x+5)(x+2))/((x-7)(x+5)(x+2))×(5(x-3)×2)/(4(x+1))

= (5(x-3)×2)/(4(x+1))

= (5×2 (x-3))/(4(x+1))

= (5(x-3))/(2(x+1))