The functions are
[tex]f(x)=x^2-6x-15[/tex][tex]g(x)=-3x-1[/tex]You have to calculate (f o g)(x), this means that you have to replace g(x) inside f(x) → f(g(x))
So for f(x) x will be g(x) as:
[tex]\begin{gathered} f(g(x))=(g(x))^2-6(g(x))-15 \\ f(g(x))=(-3x-1)^2-6(-3x-1)-15 \end{gathered}[/tex]I'll separate the composition in parts and solve them separatelly, once all terms are solved I'll add them together again:
So first solve the square of the binomial:
[tex]\begin{gathered} (-3x-1)^2 \\ (-3x-1)(-3x-1) \\ (-3x)(-3x)+(-3x)(-1)+(-1)(-3x)+(-1)(-1) \\ 9x^2+3x+3x+1 \\ 9x^2+6x+1 \end{gathered}[/tex]Next solve the second term, by applying the distributive property of multiplication:
[tex]\begin{gathered} -6(-3x-1) \\ (-6)(-3x)+(-6)(-1) \\ 18x+6 \end{gathered}[/tex]Now put both solutions toghether with the last term of the equation and order the like terms together:
[tex]\begin{gathered} f(g(x))=(-3x-1)^2-6(-3x-1)-15 \\ f(g(x))=(9x^2+6x+1)+(18x+6)-15 \\ f(g(x))=9x^2+6x+18x-15+6+1 \end{gathered}[/tex]And finally simplify the expression by solving the operations between the like terms:
[tex]\begin{gathered} f(g(x))=9x^2+(6x+18x)+(-15+6+1) \\ f(g(x))=9x^2+21x-8 \end{gathered}[/tex]
