<s>[INST] Write a polynomial equation with the following roots. A quartic funciton with roots of $-3$,$-1$ and $4$? Write a polynomial equation with the following roots. A quartic funciton with roots of $-3,-1$ and $4$?
use each root as a factor
$$f(x) = a(x + 3)(x + 1)(x - 4)^2$$
$$f(x) = a(x^2 + 4x + 3)(x^2 - 8x + 16)$$
$$f(x) = a(x^4 - 4x^3 - 13x^2 + 40x + 48)$$
$$f(5) = 16$$
$$16 = a(5^4 - 4(5^3) - 13(5^2) + 40(5) + 48 = 48a \rightarrow a = 3$$
$$f(x) = 3x^4 - 12x^3 - 39x^2 + 120x + 144$$
Is that correct? Please show another way of doing it please with your steps.
 [/INST] That's correct.  You made $4$ a double root by including two $(x-4)$ factors.  You could have equally well made $-3$ or $-1$ double roots.
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