Given:
[tex]f(x)=4x^2+x+1\text{ and }g(x)=x^2-2.[/tex]The composition is g(f(x)).
[tex]\text{ Substitute }f\mleft(x\mright)=4x^2+x+1\text{ in the g(f(x)), we get}[/tex][tex]g(f(x))=g(4x^2+x+1)[/tex][tex]Replace\text{ }x=4x^2+x+1\text{ in }g(x)=x^2-2,\text{ we get}[/tex][tex]g(f(x))=(4x^2+x+1)^2-2[/tex][tex]\text{Use (a+b+c)}^2=a^2+b^2+c^2+2ab+2bc+2ca\text{.}[/tex][tex]g(f(x))=(4x^2)^2+x^2+1^2+2(4x)(x)+2(4x)(1)+2(x)(1)-2[/tex][tex]g(f(x))=16x^4+x^2+1+8x^2+8x+2x-2[/tex]Adding like terms, which terms have the same variable with the same power.
[tex]g(f(x))=16x^4+9x^2+10x-1[/tex]Hence the answer is
[tex]g(f(x))=16x^4+9x^2+10x-1[/tex]