<s>[INST] Factoring $x^4 + 5x^3 + 4x^2 + 2x - 3$ 
So I have to factor this polynomial
  $$x^4 + 5x^3 + 4x^2 + 2x - 3$$

I got $(x^2 + 2x -3)(x^2 + 3x + 1)$
but when I multiplied it, I got a different equation:
$$x^4 + 5x^3 + 4x^2 - 7x - 3$$
I don’t really understand how to find the factors. I only found numbers that could add up to 5 and multiply to -3 but they’re wrong. What did I do wrong and what can I do to fix this?
In the format of: 
($x^2$+ __ x +__ ) ($x^2$ + __ x + __ )
or
$$(x^2 + ax + b) (x^2 + cx + d)$$
Also, how does comparing coefficients help factor this polynomial? 
 [/INST] As said in comments, there must be a typo in the question since it does not factorize.
If you write
$$(x^2 + ax + b) (x^2 + cx + d)=x^4 + 5x^3 + 4x^2 + 2x - 3$$ expanding and grouping terms, we end with
$$(b d+3)+x (a d+b c-2)+x^2 (a c+b+d-4)+x^3 (a+c-5)=0$$ and each coefficient must be zero.
So, trying, we have $d=-\frac 3b$, $c=5-a$ but the remaining equations are nightmares since we get
$$a=\frac{b (5 b-2)}{b^2+3}$$ and what is left to solve  for $b$ is
$$-27-36 b-39 b^2+47 b^3+13 b^4-4 b^5+b^6=0$$ which does not show any obvious roots (in fact, there are two real non-rational solutions).
If you had $b$, then all the other are available. 
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