Write the indicated expression as a ratio of two polynomials

Write the indicated expression as a ratio of two polynomials, assuming that

r(x)= (3x+6)/ (x^2+1) s(x)= (x^2+4)/ (4x-1)

Find s(r(x)).

r(x)= (3x+6)/ (x^2+1) ; s(x)= (x^2+4)/ (4x-1)

r(x)/s(x) = (3x+6)/ (x^2+1) / (x^2+4)/ (4x-1)

= (3x+6)(4x -1)/ (x^2+1)(x^2+4)

Find s(r(x)) Plug x= r(x) in s(x)

Numerator: (3*(3x+6)/(x^2 +1) +6 ) = (9x +18+6x^2 +6x^2 +6 )/( x^2 +1) = ( 12x^2 +9x+24)/(x^2 +1)

Denominator : ((3x+6)/ (x^2+1))^2 +1 = (9x^2 +36 +36x)/( x^4 +1+2x^2) +1

= ( (9x^2 +36 +36x)+ x^4 +1+2x^2 )/(x^4 +1+2x^2)

= ( 11x^2 +x^4 +36x +37)/(x^4 +1+2x^2)

Write Numertaor/Denominator = S(r(x)) = ( 12x^2 +9x+24)/(x^2 +1) / ( 11x^2 +x^4 +36x +37)/(x^2+1)^2

= ( 12x^2 +9x+24)(x^2+1)/( 11x^2 +x^4 +36x +37)