14\x^2-9+6\x-3=8\x+3

It is possible that this equation was meant to be:

$$\frac{14}{x^2-9}+\frac{6}{x-3}=\frac{8}{x+3}$$

in which case, multiply all terms by x² -9

$$14+\frac{6(x^2-9)}{x-3}=\frac{8(x^2-9))}{x+3}$$

Express x² - 9 as (x+3)(x-3) and cancel as appropriate

$$14+6(x+3)=8(x-3)$$

Expand bracketed terms and collect like terms on the same side

$$14+18+24=8x-6x$$

$$56=2x$$

$$x=28$$

.

$${\frac{{\mathtt{14}}}{{{\mathtt{x}}}^{{\mathtt{2}}}}}{\mathtt{\,-\,}}{\mathtt{9}}{\mathtt{\,\small\textbf+\,}}{\frac{{\mathtt{6}}}{{\mathtt{x}}}}{\mathtt{\,-\,}}{\mathtt{3}} = {\frac{{\mathtt{8}}}{{\mathtt{x}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{3}} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = {\mathtt{\,-\,}}{\frac{\left({\sqrt{{\mathtt{211}}}}{\mathtt{\,\small\textbf+\,}}{\mathtt{1}}\right)}{{\mathtt{15}}}}\\ {\mathtt{x}} = {\frac{\left({\sqrt{{\mathtt{211}}}}{\mathtt{\,-\,}}{\mathtt{1}}\right)}{{\mathtt{15}}}}\\ \end{array} \right\} \Rightarrow \left\{ \begin{array}{l}{\mathtt{x}} = -{\mathtt{1.035\: \!055\: \!936\: \!422\: \!263\: \!3}}\\ {\mathtt{x}} = {\mathtt{0.901\: \!722\: \!603\: \!088\: \!93}}\\ \end{array} \right\}$$

happy 7 is correct.

If you need to show some work:   14/x² - 9 + 6/x - 3  =  8/x + 3

Multiply each term by x² to get rid of denominators:

                (x²)(14/x²) - (x²)(9) + (x²)(6/x) - (x²)(3)  =  (x²)(8/x) + (x²)(3)

                                               14 - 9x² + 6x - 3x²  =  8x + 3x²

                                                      14 - 12x² + 6x  =  8x + 3x²

                                                       14 - 15x² - 2x  =  0 

                                                      -15x² - 2x + 14  =  0

                                                       15x² + 2x - 14  =  0

Using the quadratic equation:  x  =  [ -b ± √(b² -4ac) ] / (2a)

                             a = 15     b = 2     c = -14

                                            x  =  [ -2 ± √(2² -4·15·-14) ] / (2·15)

                                            x  =  [ -2 ± √(4 + 840) ] / (30)

                                            x  =  [ -2 ± √(844) ] / (30)

                                            x  =  [ -2 ± √4√211 ] / 30

                                            x  =  [ -2 ± 2√211 ] / 30

                                            x  =  [ -1 ± √211 ] / 15