To find the height of a rectangular prism given its volume and base area, we use the formula for volume which is [tex]\( V = B \times h \)[/tex], where [tex]\( V \)[/tex] is the volume, [tex]\( B \)[/tex] is the area of the base, and [tex]\( h \)[/tex] is the height. Given are the expressions for [tex]\( V \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[ V = 10x^3 + 46x^2 - 21x - 27 \][/tex]
[tex]\[ B = 2x^2 + 8x - 9 \][/tex]
We need to determine the height expression, [tex]\( h \)[/tex], which can be found by rearranging the formula to solve for [tex]\( h \)[/tex]:
[tex]\[ h = \frac{V}{B} \][/tex]
Substituting the given expressions:
[tex]\[ h = \frac{10x^3 + 46x^2 - 21x - 27}{2x^2 + 8x - 9} \][/tex]
We simplify this rational expression by performing polynomial division or factorization. The resulting expression for the height [tex]\( h \)[/tex] simplifies to:
[tex]\[ h = 5x + 3 \][/tex]
Thus, the expression that represents the height of the prism is:
[tex]\[ 5x + 3 \][/tex]
So out of the given choices, the correct answer is:
[tex]\[ \boxed{5x + 3} \][/tex]
