<s>[INST] Factoring $1 - y-x^2-y^2-yx^2+y^3$ I was struggling to factorize this expression, I need it cleaned to bracket form but I was struggling to understand the process to get the answer? Is there a special formula or technique for factorizing these sorts of expressions?
$$1 - y-x^2-y^2-yx^2+y^3$$
Answer: $$ \ ( 1+y)[(1-y)^2-x^2]   $$
 [/INST] Here is how I factored it:
$$1-y-x^2-y^2-yx^2+y^3=$$
$$(1-y^2)-(y-y^3)-(x^2+yx^2)=$$
$$(1-y^2)-y(1-y^2)-x^2(1+y)=$$
$$(1-y^2)(1-y)-x^2(1+y)=$$
$$(1-y)(1+y)(1-y)-x^2(1+y)=$$
$$(1+y)(1-y)^2-(1+y)x^2=$$
$$\boxed{(1+y)[(1-y)^2-x^2]}$$
You can factorize it further:
$$(1+y)[(1-y)^2-x^2]=$$
$$(1+y)[(1-y-x)(1-y+x)]=$$
$$\boxed{(1+y)(1-y-x)(1-y+x)}$$
We can expand $(1+y)(1-y-x)(1-y+x)$ to make sure we got it:
$$\require{cancel} (1+y)(1-y-x)(1-y+x)=$$
$$(1\cancel{-y}-x\cancel{+y}-y^2-xy)(1-y+x)=$$
$$(1-x-y^2-xy)(1-y+x)=$$
$$(1-x-y^2-xy)-(y-xy-y^3-xy^2)+(x-x^2-xy^2-x^2y)=$$
$$1\cancel{-x}-y^2\cancel{-xy}-y\cancel{+xy}+y^3\cancel{+xy^2}\cancel{+x}-x^2-\cancel{xy^2}-x^2y=$$
$$1-y^2-y+y^3-x^2-x^2y=$$
$$\boxed{1-y-x^2-y^2-yx^2+y^3}$$
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