9 1 12x3x 275x2x SolutionWe have 514x3x212x3x297x2x22715x2x2

Solution

We have [(5-14x-3x²)/(1-2x-3x²)]*[(9+7x-2x²)/(27-15x+2x²)] = [(5/3-14x/3-x²)/(1/3-2x/3-x²)] *[(9/2+7x/2-x²)/ (27/2-15x/2+x²)] = [-(x+5)(x-1/3)/-(x+1)(x-1/3)]*[-(x+1)(x-9/2)/(x-6)(x-9/2)] =-[(x-1/3)/(x-1/3)]*[(x+5)/(x-6)]* [(x-9/2)/(x-9/2)]*[(x+1)/(x+1)] =-(x+5)/(x-6) = (x+5)/(6-x).

Note: We have used the quadratic formula for factorization e.g. (5-14x-3x²)= -(3x²+14x -5). Now as per the quadratic formula, the roots of x²+14x/3 -5/3 are [-14/3±(14²/3²-4*1*(-5/3)]/2*1 or, [-14/3±( 196/9+60/9)]/2 or, [-14/3±256/9]/2 or, (-14/3± 16/3)/2 i.e. 2/6 = 1/3 and -30/6 =   -5. Hence, 5/3-14x/3-x² = -(x+5)(x-1/3). The same process has been used for factorizing the other polynomials.