Question

F(x) = 3x + 2

g(x) = x^2 + 1
find gf(x) in the form ax^2 + bx + c

please add an explanation because i’ve seen it being solved i just do not understand some steps

like, where does 12x come from

Answer 1

g(f(x)) = 9x² + 12x + 5

Explanation:

g(f(x))

= g(3x + 2) ← substitute x = 3x + 2 into g(x)

= (3x + 2)² + 1

to expand (3x + 2)² = (3x + 2)(3x + 2)

each term in the second factor is multiplied by each term in the first factor

3x(3x + 2) + 2(3x + 2) ← distribute both parenthesis

= 9x² + 6x + 6x + 4 ← collect like terms

= 9x² + 12x + 4

then

g(f(x)) = (3x + 2)² + 1 = 9x² + 12x + 4 + 1 = 9x² + 12x + 5