Question

complete the lesson reflection above by circling your current understanding of the learning goals.
factor the polynomial.

1. $8k^2 + 80k + 200$

modeling with mathematics

1. the polynomial represents the area (in square feet) of the square playground.

a. write a polynomial that represents the side length of the playground. hint: factor the polynomial to find the side length.
b. write an expression for the perimeter of the playground. hint: the perimeter of any polygon is the sum of the lengths of all the sides.
(image: a square playground with $a = x^2 - 30x + 225$)

Explanation:

For problem 38: $8k^2 + 80k + 200$

Step1: Factor out GCF

Factor out 8 from all terms.
$8(k^2 + 10k + 25)$

Step2: Factor quadratic trinomial

Recognize $k^2 +10k +25$ as a perfect square.
$k^2 +10k +25=(k+5)^2$

Step3: Combine factors

Substitute the factored quadratic back.
$8(k+5)^2$

Step1: Identify area polynomial

Area of square: $A=x^2-30x+225$

Step2: Factor perfect square trinomial

Rewrite as square of binomial.
$x^2-30x+225=(x-15)^2$

Step3: Relate to side length

Side length of square is the square root of area.
$\sqrt{(x-15)^2}=x-15$

Step1: Recall square perimeter formula

Perimeter $P=4\times$ side length

Step2: Substitute side length

Use side length from part a: $x-15$.
$P=4(x-15)$

Step3: Simplify the expression

Distribute the 4 to terms inside.
$4x-60$

Answer:

$8(k+5)^2$

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For problem 26:

Part a