How to factor this polynominal?

[hkus10]

1) (x-10)[(x+4)(x+1) - 24] - 3[(-11x - 11) + 24] + 8[-21 + 3x]

what I get is (x-10)(x^2+5x-20) + 57x-207
The reason that I do not combine them is because I think it is much more difficult to deal with x^3?
What should I do here?

 

[Dick]

It looks to me like you should just multiply the whole thing out and then try to factor it. Once you've done that you can use the rational roots theorem to find possible factors. There aren't very many.

 

[SammyS]

1) (x-10)[(x+4)(x+1) - 24] - 3[(-11x - 11) + 24] + 8[-21 + 3x]

what I get is (x-10)(x^2+5x-20) + 57x-207
The reason that I do not combine them is because I think it is much more difficult to deal with x^3?
What should I do here?

There are lots of (x+1)s & 24s appears here & there ...

Try something like:

$$(x-10)[(x+4)(x+1) - 24] - 3[(-11x - 11) + 24] + 8[-21 + 3x]$$

$$=<br /> \left((x+1)-11\right)\left[ \left((x+1)+3\right)(x+1)-24\right]+3\left(11(x+1)-24\right)+8\left[3(x+1)-24\right]$$​

Now there are more (x+1)s & 24 plus more 3s & 11s. Multiply these out. Don't do too much arithmetic.

$$=\left((x+1)-11\right)\left[(x+1)^2+3(x+1)-24\right]+33(x+1)-3\cdot24+24(x+1)-8\cdot24$$

$$=(x+1)^3+3(x+1)^2-24(x+1)-11(x+1)^2-33(x+1)+11\cdot24\ \ +\ 33(x+1)+24(x+1)-11\cdot24$$​

Lots of stuff cancels out at this point: 11*24, 33(x+1), 24(x+1) .

What's left?