Question

lesson 18.2 checkpoint□ once you have completed the above problems and checked your solutions, complete the lesson checkpoint below.□ complete the lesson reflection above by circling your current understanding of the learning goal.find each product.1. $(4x + 2)(x - 2)$2. $(4x^2 - 4)(2x + 1)$3. $(x^2 + 3)(x^3 - x^2 + 4)$write a simplified polynomial expression for the situation, then evaluate.4. if the side length of a square can be represented by $2x - 5$ meters, write an expression that can be used to represent the area of the square. simplify the expression as much as possible.5. what is the area of the square when $x = 7?$

Explanation:

Step1: Apply distributive property (FOIL)

$(4x+2)(x-2) = 4x\cdot x + 4x\cdot(-2) + 2\cdot x + 2\cdot(-2)$

Step2: Simplify each term

$=4x^2 -8x +2x -4$

Step3: Combine like terms

$=4x^2 -6x -4$

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Step1: Apply distributive property

$(4x^2-4)(2x+1) = 4x^2\cdot2x + 4x^2\cdot1 -4\cdot2x -4\cdot1$

Step2: Simplify each term

$=8x^3 +4x^2 -8x -4$

Step3: Combine like terms (none left)

$=8x^3 +4x^2 -8x -4$

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Step1: Distribute each term

$(x^2+3)(x^3-x^2+4) = x^2\cdot x^3 + x^2\cdot(-x^2) + x^2\cdot4 + 3\cdot x^3 + 3\cdot(-x^2) + 3\cdot4$

Step2: Simplify each term

$=x^5 -x^4 +4x^2 +3x^3 -3x^2 +12$

Step3: Combine like terms

$=x^5 -x^4 +3x^3 +x^2 +12$

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Step1: Use square area formula

$\text{Area}=(2x-5)^2$

Step2: Expand the square

$=(2x)^2 - 2\cdot2x\cdot5 +5^2$

Step3: Simplify each term

$=4x^2 -20x +25$

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Step1: Substitute $x=7$ into area formula

$\text{Area}=4(7)^2 -20(7)+25$

Step2: Calculate each term

$=4\cdot49 -140 +25 = 196 -140 +25$

Step3: Compute final value

$=81$

Answer:

1. $\boldsymbol{4x^2 -6x -4}$
2. $\boldsymbol{8x^3 +4x^2 -8x -4}$
3. $\boldsymbol{x^5 -x^4 +3x^3 +x^2 +12}$
4. $\boldsymbol{4x^2 -20x +25}$
5. $\boldsymbol{81}$ square meters