Area of a rectangle is n^2 -4n -5, where n is a positive number. If one side has length n+1, give the second side in terms of n. Working pls?

Area of the rectangle = (Length of the first side) × (Length of the second side)

Length of the second side
= (Area of the rectangle) / (Length of the first side)
= (n² - 4n - 5) / (n + 1)
= (n + 1)(n - 5) / (n + 1)
= n - 5

n^2 -4n -5 = n^2 +n -5n -5 = n(n+1)-5(n+1) = (n-5)(n+1)

The other side is n-5.

n + 1 = n - (- 1)

Using synthetic division :-

- 1 I 1_______- 4______-5
__ I________- 1_______5
__I 1_______ - 5______0

n - 5 is length of second side

Check
[ n - 5 ] [ n + 1 ] = n² - 4n - 5

2nd side is: n^2 -4n -5/n+1 = n-5
Check for area: (n+1)(n-5) = n^2 -4n -5

ℓ: length of the rectangle → (n + 1)

ω: width of the rectangle



The area of a rectangle is

area = ℓ * ω → given that it’s: n² - 4n - 5

ℓ * ω = n² - 4n - 5 → given that: ℓ = n + 1

(n + 1) * ω = n² - 4n - 5 → suppose that: ω = n + a

(n + 1).(n + a) = n² - 4n - 5 → you expand

n² + an + n + a = n² - 4n - 5

n² + n.(a + 1) + a = n² - 4n - 5 → you compare both sides

a = - 5

a + 1 = - 4 → a = - 5 of course because above


Recall: ω = n + a

→ ω = n - 5

Area of a rectangle:
LW = n^2 - 4n - 5
L = n + 1
W = n - 5

A = LW
= n² - 4n - 5
= (n-5)(n+1)
If L = n+1, then W = n-5

Ans: n-5

n-5