Question

remote learning day!
card one: ((x + 2)(x + 4))
card two: ((y + 5)(y + 1))
card three: ((4x + y)(7x - 2y))
card four: ((2r + s)(2r - s))
card five: ((a + 3b)^2)
card six: ((3x + 1)(5x^2 + 2x - 6))
card seven: ((2x + 2)(4x^2 - 3x - 6))
card eight: ((5pq^2 - 2p^4q + pq - 9) + (10pq^2 + p^4q - 13pq))
card nine: find the area of a triangle with a base of (3xy^2) and a height of (2x^2).
card ten: find the area of a square with a side of ((6x + 2)).
card eleven: what is the area of the figure? (image of a rectangle and square)
card twelve: ((5a - 5b + 1) - (3a + 2b))
card thirteen: (2x^2(x^2 - 4xy - y^2))
card fourteen: ((5x + 2)(4x - 3))
card fifteen: ((x - 1)(2x^2 - x + 5))
card sixteen: ((6xy^2 + 8x^2y + xy^2 - 2) - (10xy^2 + x^2y - 20xy))
write your final answer inside each box.
you can do work on the back!

Explanation:

Card One: Step1: Apply FOIL method

$(x+2)(x+4) = x^2 +4x +2x +8$

Card One: Step2: Combine like terms

$x^2 + (4x+2x) +8 = x^2+6x+8$

Card Two: Step1: Apply FOIL method

$(y+5)(y+1) = y^2 +y +5y +5$

Card Two: Step2: Combine like terms

$y^2 + (y+5y) +5 = y^2+6y+5$

Card Three: Step1: Apply FOIL method

$(4x+y)(7x-2y) = 28x^2 -8xy +7xy -2y^2$

Card Three: Step2: Combine like terms

$28x^2 + (-8xy+7xy) -2y^2 = 28x^2 -xy -2y^2$

Card Four: Step1: Use difference of squares

$(2r+s)(2r-s) = (2r)^2 - s^2$

Card Four: Step2: Simplify exponents

$4r^2 - s^2$

Card Five: Step1: Expand perfect square

$(a+3b)^2 = a^2 + 2(a)(3b) + (3b)^2$

Card Five: Step2: Simplify terms

$a^2 +6ab +9b^2$

Card Six: Step1: Distribute $3x$ and $1$

$(3x+1)(5x^2+2x-6) = 3x(5x^2+2x-6) +1(5x^2+2x-6)$

Card Six: Step2: Compute each product

$15x^3 +6x^2 -18x +5x^2 +2x -6$

Card Six: Step3: Combine like terms

$15x^3 + (6x^2+5x^2) + (-18x+2x) -6 = 15x^3+11x^2-16x-6$

Card Seven: Step1: Distribute $2x$ and $2$

$(2x+2)(4x^2-3x-6) = 2x(4x^2-3x-6) +2(4x^2-3x-6)$

Card Seven: Step2: Compute each product

$8x^3 -6x^2 -12x +8x^2 -6x -12$

Card Seven: Step3: Combine like terms

$8x^3 + (-6x^2+8x^2) + (-12x-6x) -12 = 8x^3+2x^2-18x-12$

Card Eight: Step1: Remove parentheses

$(3pq^2 -2p^2q +pq -9) + (10pq^2 +p^2q -13pq) = 3pq^2 -2p^2q +pq -9 +10pq^2 +p^2q -13pq$

Card Eight: Step2: Combine like terms

$(3pq^2+10pq^2) + (-2p^2q+p^2q) + (pq-13pq) -9 = 13pq^2 -p^2q -12pq -9$

Card Nine: Step1: Use triangle area formula

$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height}$

Card Nine: Step2: Substitute given values

$\text{Area} = \frac{1}{2} \times 3xy^2 \times 2x^3$

Card Nine: Step3: Multiply coefficients and variables

$\frac{1}{2} \times 3 \times 2 \times x \times x^3 \times y^2 = 3x^4y^2$

Card Ten: Step1: Use square area formula

$\text{Area} = (\text{side})^2$

Card Ten: Step2: Substitute side length

$\text{Area} = (6x+2)^2$

Card Ten: Step3: Expand perfect square

$(6x)^2 + 2(6x)(2) + 2^2 = 36x^2 +24x +4$

Card Eleven: Step1: Split figure into 2 rectangles

Let the large square have side $x$, small square cut out have side $x-2$ (from diagram). Area = $x^2 - (x-2)^2$

Card Eleven: Step2: Expand both squares

$x^2 - (x^2 -4x +4)$

Card Eleven: Step3: Simplify expression

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

Card Twelve: Step1: Remove parentheses

$(5a-5b+1)-(3a+2b) = 5a-5b+1-3a-2b$

Card Twelve: Step2: Combine like terms

$(5a-3a) + (-5b-2b) +1 = 2a -7b +1$

Card Thirteen: Step1: Distribute $2x^2$

$2x^2(x^2-4xy-y^2) = 2x^2(x^2) +2x^2(-4xy) +2x^2(-y^2)$

Card Thirteen: Step2: Simplify each term

$2x^4 -8x^3y -2x^2y^2$

Card Fourteen: Step1: Apply FOIL method

$(5x+2)(4x-3) = 20x^2 -15x +8x -6$

Card Fourteen: Step2: Combine like terms

$20x^2 + (-15x+8x) -6 = 20x^2 -7x -6$

Card Fifteen: Step1: Distribute $x$ and $-1$

$(x-1)(2x^2-x+5) = x(2x^2-x+5) -1(2x^2-x+5)$

Card Fifteen: Step2: Compute each product

$2x^3 -x^2 +5x -2x^2 +x -5$

Card Fifteen: Step3: Combine like terms

$2x^3 + (-x^2-2x^2) + (5x+x) -5 = 2x^3-3x^2+6x-5$

Card Sixteen: Step1: Remove parentheses

$(6xy^2+8x^2y+xy-2)-(10xy^2+x^2y+3xy) = 6xy^2+8x^2y+xy-2-10xy^2-x^2y-3xy$

Card Sixteen: Step2: Combine like terms

$(6xy^2-10xy^2) + (8x^2y-x^2y) + (xy-3xy) -2 = -4xy^2 +7x^2y -2xy -2$

Answer:

Card One: $\boldsymbol{x^2+6x+8}$
Card Two: $\boldsymbol{y^2+6y+5}$
Card Three: $\boldsymbol{28x^2 -xy -2y^2}$
Card Four: $\boldsymbol{4r^2 - s^2}$
Card Five: $\boldsymbol{a^2 +6ab +9b^2}$
Card Six: $\boldsymbol{15x^3+11x^2-16x-6}$
Card Seven: $\boldsymbol{8x^3+2x^2-18x-12}$
Card Eight: $\boldsymbol{13pq^2 -p^2q -12pq -9}$
Card Nine: $\boldsymbol{3x^4y^2}$
Card Ten: $\boldsymbol{36x^2 +24x +4}$
Card Eleven: $\boldsymbol{4x -4}$
Card Twelve: $\boldsymbol{2a -7b +1}$
Card Thirteen: $\boldsymbol{2x^4 -8x^3y -2x^2y^2}$
Card Fourteen: $\boldsymbol{20x^2 -7x -6}$
Card Fifteen: $\boldsymbol{2x^3-3x^2+6x-5}$
Card Sixteen: $\boldsymbol{-4xy^2 +7x^2y -2xy -2}$