To find \((f \cdot g)(x)\), we need to multiply the functions \(f(x)\) and \(g(x)\) together.
Given:
- \(f(x) = 2x^2 - 3x\)
- \(g(x) = x^2 - 6\)
We will calculate the product \(f(x) \cdot g(x)\):
\[ (f \cdot g)(x) = f(x) \cdot g(x) = (2x^2 - 3x)(x^2 - 6) \]
Now, we can use the distributive property (or FOIL) to multiply these two polynomials:
\[ = 2x^2(x^2) + 2x^2(-6) + (-3x)(x^2) + (-3x)(-6) \]
Calculating each term:
- \(2x^2 \cdot x^2 = 2x^4\)
- \(2x^2 \cdot -6 = -12x^2\)
- \(-3x \cdot x^2 = -3x^3\)
- \(-3x \cdot -6 = 18x\)
Now, combine all these results:
\[ (f \cdot g)(x) = 2x^4 - 3x^3 - 12x^2 + 18x \]
Thus, the product \((f \cdot g)(x)\) is:
\[ \boxed{2x^4 - 3x^3 - 12x^2 + 18x} \]
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