Given the expression:
[tex](x+3)-\lbrack(x+2)(x^3-1)\rbrack[/tex]
You can simplify it as follows:
1. Apply the FOIL Method to multiply the binomials inside the square brackets. This method states that:
[tex](a+b)(c+d)=ac+ad+bc+bd[/tex]
You need to remember the Sign Rules for Multiplication:
[tex]\begin{gathered} +\cdot+=+ \\ -\cdot-=+ \\ +\cdot-=- \\ -\cdot+=- \end{gathered}[/tex]
It is important to remember that, according to the Product of Powers Property, you need to add the exponents when you multiply powers with the same base.
Then, you get:
[tex]=(x+3)-\lbrack(x)(x^3)-(x)(1)+(2)(x^3)-(2)(1)\rbrack[/tex][tex]=(x+3)-\lbrack x^4-x+2x^3-2\rbrack[/tex]
2. Distribute the negative sign:
[tex]=(x+3)-x^4+x-2x^3+2[/tex]
3. Add the like terms:
[tex]=x+3-x^4+x-2x^3+2[/tex][tex]=-x^4-2x^3+2x+5[/tex]
Hence, the answer is: Option A.
