The edges of a triangular stage prop are (3x - 4) feet, (x^2 - 1) feet and (2x^2 - 15) feet. Find the perimeter when x = 4.

Step-by-step solution

1. 1

   Recall what perimeter means for a polygon. It is the total distance around the shape - the sum of the three edge lengths, not a product and not an area:

   $P=(3x-4)+(x^{2}-1)+(2x^{2}-15)$

2. 2

   Combine like terms by degree. Group the $x^2$ terms, the $x$ terms and the constants separately:

   $x^{2}+2x^{2}=3x^{2},\qquad 3x,\qquad -4-1-15=-20$

   $P=3x^{2}+3x-20$

   Simplifying first pays off: one substitution now instead of three.

3. 3

   Substitute $x=4$.

   $P=3(4)^{2}+3(4)-20=3(16)+12-20$

4. 4

   Evaluate.

   $48+12-20=40\ \text{feet}$

5. 5

   Check by evaluating the three sides separately. $3(4)-4=8$, $4^{2}-1=15$, $2(4)^{2}-15=17$; and $8+15+17=40$ ✓. The two routes agree, so the like-term collection was correct.

6. 6

   Confirm the sides can form a triangle. The triangle inequality needs the two shorter sides to exceed the longest: $8+15=23>17$ ✓, so $8,15,17$ is a genuine triangle and the answer of $40$ feet is physically meaningful.