To find \((f \cdot g)(x)\), we need to multiply the functions \(f(x)\) and \(g(x)\):
Given: \[ f(x) = 2x^2 - 3x \] \[ g(x) = x^2 - 6 \]
The product \((f \cdot g)(x)\) is computed as follows: \[ (f \cdot g)(x) = f(x) \cdot g(x) = (2x^2 - 3x)(x^2 - 6) \]
Now, we will use the distributive property (also known as the FOIL method) to multiply these two polynomials:
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Multiply \(2x^2\) by each term in \(g(x)\): \[ 2x^2 \cdot x^2 = 2x^4 \] \[ 2x^2 \cdot (-6) = -12x^2 \]
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Multiply \(-3x\) by each term in \(g(x)\): \[ -3x \cdot x^2 = -3x^3 \] \[ -3x \cdot (-6) = 18x \]
Now, combine all these terms together: \[ (f \cdot g)(x) = 2x^4 - 3x^3 - 12x^2 + 18x \]
The final result is: \[ 2x^4 - 3x^3 - 12x^2 + 18x \]
Thus, the answer is: \[ \boxed{2x^4 - 3x^3 - 12x^2 + 18x} \]