To find the expression that is equivalent to the product of \((2h^2 + 9h - 5)\) and \((3h + 6)\), we will use the distributive property (also known as the FOIL method for binomials) to expand the product.
The expression can be written as:
\[ (2h^2 + 9h - 5)(3h + 6) \]
We will distribute each term in the first polynomial to each term in the second polynomial.
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Distributing \(2h^2\):
- \(2h^2 \cdot 3h = 6h^3\)
- \(2h^2 \cdot 6 = 12h^2\)
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Distributing \(9h\):
- \(9h \cdot 3h = 27h^2\)
- \(9h \cdot 6 = 54h\)
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Distributing \(-5\):
- \(-5 \cdot 3h = -15h\)
- \(-5 \cdot 6 = -30\)
Now, we will combine all the terms we obtained:
\[ 6h^3 + 12h^2 + 27h^2 + 54h - 15h - 30 \]
Next, we gather like terms:
- \(6h^3\)
- For the \(h^2\) terms: \(12h^2 + 27h^2 = 39h^2\)
- For the \(h\) terms: \(54h - 15h = 39h\)
So the final expression becomes:
\[ 6h^3 + 39h^2 + 39h - 30 \]
Thus, the expression equivalent to the product of \((2h^2 + 9h - 5)\) and \((3h + 6)\) is:
\[ \boxed{6h^3 + 39h^2 + 39h - 30} \]