Answer :
The expression for length of rectangle is (x - 16) units
Solution:
Given that area of a rectangle is represented by [tex]x^2-14x-32[/tex] square units
width of the rectangle is represented by (x+2) units
To find: length of the rectangle
The area of rectangle is given as:
[tex]\text { area of rectangle }=\text { length } \times \text { width }[/tex]
Substituting the values in formula, we get
[tex]x^{2}-14x-32=\text{length} \times(x+2)[/tex] ---- eqn 1
Let us first factorise the L.H.S of eqn 1
[tex]x^{2}-14x-32[/tex]
Break the expression into groups
[tex]=\left(x^2+2x\right)+\left(-16x-32\right)[/tex]
[tex]\mathrm{Factor\:out\:}x\mathrm{\:from\:}x^2+2x\mathrm{:\quad }x\left(x+2\right)[/tex]
[tex]\mathrm{Factor\:out\:}-16\mathrm{\:from\:}-16x-32\mathrm{:\quad }-16\left(x+2\right)[/tex]
[tex]=x\left(x+2\right)-16\left(x+2\right)[/tex]
[tex]\mathrm{Factor\:out\:common\:term\:}x+2[/tex]
[tex]=\left(x+2\right)\left(x-16\right)[/tex]
Now substitute the above value in eqn 1
[tex](x + 2)(x - 16) = length \times (x + 2)[/tex]
[tex]length = \frac{(x + 2)(x - 16)}{x + 2}[/tex]
Length = x - 16
Thus the expression for length of rectangle is (x - 16) units