To find \((mn)(x)\), we need to first compute the product of the functions \(m(x)\) and \(n(x)\):
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Identify the functions: \[ m(x) = x^2 + 3 \] \[ n(x) = 5x + 9 \]
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Compute \((mn)(x) = m(x) \cdot n(x)\): \[ (mn)(x) = (x^2 + 3)(5x + 9) \]
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Use the distributive property (also known as the FOIL method for binomials) to expand the product: \[ = x^2 \cdot 5x + x^2 \cdot 9 + 3 \cdot 5x + 3 \cdot 9 \] \[ = 5x^3 + 9x^2 + 15x + 27 \]
So, we find that \((mn)(x) = 5x^3 + 9x^2 + 15x + 27\).
Now let's compare it with the given expressions:
- 5x^3 + 9x^2 + 15x + 27 (This is our result)
- 25x^2 + 90x + 84
- x^2 + 5x + 12
- 5x^2 + 24
The expression equivalent to \((mn)(x)\) is: \[ \boxed{5x^3 + 9x^2 + 15x + 27} \]
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