Function tranlsation $g(x) = f(x) + 15$

Question

I can't seem to work this answer out when practicing for exams.

Here's the question:

You are given that $f(x) = (2x - 3)(x + 2)(x + 4) \cdots$

From this I know $f(x)$'s roots: $\frac{3}{2}$, $-2$ and $-4$; and that it expands to: $2x^3 + 7x^2 − 10x − 24$. I use these facts to answer some related questions.

I am then asked:

You are also given that $g(x) = f(x) + 15$, show that $g(x) = 2x^3 + 9x^2 -2x - 9$.

I originally expanded $f(x)$ but this quickly showed it's not correct. Help?

Answer 1

The first three terms of your expansion of $f(x)$ must be the same as those for $g(x)$ since $g(x)$ is just a vertical translation of $f(x)$.

Answer 2

\begin{align*} f(x)&=(2x−3)(x+2)(x+4) \\&= 2x(x+2)(x+4)-3(x+2)(x+4) \\&= 2x^2(x+4)+4x(x+4)-3x(x+4)-6(x+4) \\&= 2x^3+8x^2+4x^2+16x-3x^2-12x-6x-24 \\&= 2x^3+(8+4-3)x^2+(16-12-6)x-24 \\&= 2x^3+9x^2-2x-24 \end{align*}