Solve the following quadratic equations by factorization:

Solution:

We have been given

$\frac{x+3}{x-2}-\frac{1-x}{x}=\frac{17}{4}$

$4\left(x^{2}+3 x-x+x^{2}+2-2 x\right)=17\left(x^{2}-2 x\right)$

$9 x^{2}-34 x-8=0$

$9 x^{2}-36 x+2 x-8=0$

$9 x(x-4)+2(x-4)=0$

$(9 x+2)(x-4)=0$

Therefore,

$9 x+2=0$

$9 x=-2$

$x=\frac{-2}{9}$

or,

$x-4=0$

$x=4$

Hence, $x=\frac{-2}{9}$ or $x=4$.