Question 604393
{{{2x^3}}} is a monomial, a group of one or more numbers and/or symbols multiplied together, with no plus or minus signs. Monomials can be as simple as {{{x}}} or {{{2}}}, or not so simple as {{{24x^5y^2z^3}}}.
 
{{{y+1}}} and {{{4x^2-9y^4}}} are both binomials because each has 2 groups of symbols and/or numbers (2 monomials) separated by plus and/or minus signs. The "bi" part means 2, as in bicycle.
 
{{{x^2-5x+6}}} is a trinomial because there are 3 groups of symbols and numbers separated by plus and/or minus signs. The "tri" part means 3, as in tricycle.
The binomials {{{x-3}}} and {{{x-2}}} can be multiplied together and the result is the trinomial {{{x^2-5x+6}}} :
{{{(x-2)(x-3)=x^2-3x-2x+6=x^2-5x+6}}}
 
To start, the expression
{{{x^4(y+1)+2(y+1)x^3+x^2(y+1)}}} can be simplified by
"taking out" the common factor {{{(y+1)}}} .
{{{x^4(y+1)+2(y+1)x^3+x^2(y+1)=(x^4+2x^3+x^2)(y+1)}}}
 
The trinomial {{{x^4+2x^3+x^2}}} in that expression can be factored further.
To begin, we can "take out" the common factor {{{x^2}}}
{{{x^4+2x^3+x^2=x^2(x^2+2x+1)}}}
 
Then, the trinomial {{{x^2+2x+1}}} can be factored further. It happens to be the square of the binomial {{{x+1}}}
{{{(x+1)^2=x^2+2x+1}}}
 
So in 3 steps, we can factor {{{x^4(y+1)+2(y+1)x^3+x^2(y+1)}}} completely:
{{{x^4(y+1)+2(y+1)x^3+x^2(y+1)=(x^4+2x^3+x^2)(y+1)=x^2(x^2+2x+1)(y+1)=x^2(x+1)^2(y+1)}}}