Answer:
D
Step-by-step explanation:
The area of a rectangle is given by the formula:
[tex]A=\ell w[/tex]
So, we are given that the area is:
[tex]x^3-5x^2+3x-15[/tex]
And the width is:
[tex]x^2+3[/tex]
And we want to find the length. To do so, first substitute the expressions into the equation:
[tex]x^3-5x^2+3x-15=(x^2+3)\ell[/tex]
Thus, to find the length, divide by (x²+3):
[tex]\displaystyle \ell = \frac{x^3-5x^2+3x-15}{x^2+3}[/tex]
We can factor the numerator:
[tex]x^3-5x^2+3x-15[/tex]
From the first two terms, factor out a x².
From the third and fourth terms, factor out a 3:
[tex]=x^2(x-5)+3(x-5)[/tex]
Combine:
[tex]=(x^2+3)(x-5)[/tex]
Putting this back:
[tex]\displaystyle \ell = \frac{(x^2+3)(x-5)}{x^2+3}[/tex]
Cancel:
[tex]\ell =x-5[/tex]
Hence, our answer is D.
