<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] $(x^2+ax+b)^2=x^4+2ax^3+a^2x^2+2bx^2+2abx+b^2$
we find:
$a=\frac{5}{2}$
$b=\frac{-9}{8}$
and we get following polynomial with these values:
$x^4+5x^3+4x^2-\frac{45}{8}x+\frac{81}{64}$
We compare this with given polynomial; if: 
$x^4+5x^3+4x^2+2x-3=0$
then:
$(x^2+\frac{5}{4}x-\frac{9}{8})^2=\frac{273}{64}-\frac{61}{8}x$
This equation has two real roots ; $x_1≈-4.202$ and $x_2≈0.53$ and two complex roots ; $x_3≈ -0.66-0.95 i$ and $x_3≈ -0.66+0.95 i$, so the factorized form can be:
$(x-0.52)(x+4.2)(x+0.66+095 i)(x-0.66+095 i)$
You can get the same result from Wolfram.
</s>