Answer:
[tex]h = x^2+3x-5[/tex]
Step-by-step explanation:
Given
[tex]V = x^4 + 11x^3 + 34x^2 + 5x - 75[/tex]
[tex]L = x + 5[/tex]
[tex]W = x + 3[/tex]
Required
Determine the height (h) of the prism
Volume is calculated as:
[tex]V =lwh[/tex]
Substitute values for V, l and w
[tex]x^4 + 11x^3 + 34x^2 + 5x - 75 = (x + 5) * (x + 3) * h[/tex]
Factorize the expression on the left-hand side
[tex](x+3)(x^3+8x^2+10x-25)= (x + 5) * (x + 3) * h[/tex]
Further, factorize:
[tex](x+3)(x+5)(x^2+3x-5)= (x + 5) * (x + 3) * h[/tex]
Divide both sides by (x+3)(x+5)
[tex]\frac{(x+3)(x+5)(x^2+3x-5)}{(x+3)(x+5)}= \frac{(x + 5) * (x + 3) * h}{(x+3)(x+5)}[/tex]
[tex]\frac{(x+3)(x+5)(x^2+3x-5)}{(x+3)(x+5)}= h[/tex]
[tex](x^2+3x-5)= h[/tex]
[tex]x^2+3x-5= h[/tex]
[tex]h = x^2+3x-5[/tex]
The height of the prism is [tex]x^2+3x-5[/tex]
