How would I determine the value of x where the graph of f has a horizontal tangent line.?

Question

f(x) = (x^2+3)(x^2+4x).

Answer 1

At a horizontal tangent, the slope is zero.
Slope measures rate of change.
You can use calculus to differentiate the function f(x).
Once you do this, set df/dx to zero and solve for x.

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Answer 2

Seting the first derivative of your function = 0 gives
x³ + 3x² + 1.5x + 3 = 0

Solve this cubic for x and you will have the answer