e. To make factoring easier, rearrange the 25 - 20x + 4x2 .
6x2−11x−106x2−5x−6⋅6−4x4x2−20x+25
Now we need to factor all the numerators and denominators. Start by splitting their middle terms.
= 6x2+4x−15x−106x2−9x+4x−6⋅6−4x4x2−10x−10x+25
Now factor by grouping.
= 2x(3x+2)−5(3x+2)3x(2x−3)+2(2x−3)⋅6−4x2x(2x−5)−5(2x−5)
= (3x+2)(2x−5)(2x−3)(3x+2)⋅−2(2x−3)(2x−5)(2x−5)
Multiply the fractions together.
= (3x+2)(2x−5)(−2)(2x−3)(2x−3)(3x+2)(2x−5)(2x−5)
Cancel the common terms.
= (3x+2)(2x−5)(−2)(2x−3)(2x−3)(3x+2)(2x−5)(2x−5)
= (−2)(2x−5)
= - 22x−5
And x ≠ 3/2 or -2/3 since these cause a zero in the denominator of the original expression.