rational expressions: find (3x)/(x + 3) + (x + 2)/(x - 7)

I am stuck in the middle of my problem. Was hoping that someone could check my work.

Find: (3x)/(x + 3) + (x + 2)/(x - 7)

My work:

. . .[3x(x - 7) + (x + 2)(x + 3)] / [(x + 3)(x - 7)]

. . .(3x^2 - 21x + x^2 + 3x + 2x + 6) / [(x + 3)(x - 7)]

. . .(4x^2 - 16x + 6) / [(x + 3)(x - 7)]

I need to reduce the top line to lowest terms.

. . .2(x^2 - 8x + 3) = 2(2x - 3)(x - 1)

...or (2x-?) (2x - ?)

This is where I am mixed up and could use some guidance.

Still trying to figure this out. Thanks in advance.

You got 'er whittled down to:

\(\displaystyle \L\\\frac{2(2x^{2}-8x+3)}{(x+3)(x-7)}=0\)

Now, I believe the kibosh has been put on factoring the numerator any further.

Use the quadratic formula on the numerator. Maybe complete the square.