Multiply and then simplify by collecting like terms: (a) (2𝑥 + 1)(𝑥 + 3) (b) (2 − 𝑦)(2 − 𝑦) (c) (𝑦3 + 𝑦)(𝑦2 + 1) (d) (𝑥 + 1)(𝑥 + 1) (e) (𝑥 + 𝑦)(𝑥 − 𝑦

(a) Multiply each term in the first binomial by each term in the second binomial:
2x \cdot x = 2x^2
2x \cdot 3 = 6x
1 \cdot x = x
1 \cdot 3 = 3
Combine like terms:
2x^2 + 6x + x + 3 = 2x^2 + 7x + 3

(b) Expand the expression by multiplying:
(2 - y)(2 - y) = (2 - y)^2 = 2^2 - 2\cdot2\cdot y + y^2 = 4 - 4y + y^2
Reorder:
y^2 - 4y + 4

(c) Multiply each term:
y^3 \cdot y^2 = y^5
y^3 \cdot 1 = y^3
y \cdot y^2 = y^3
y \cdot 1 = y
Combine like terms:
y^5 + y^3 + y^3 + y = y^5 + 2y^3 + y

(d) Expand the square of a binomial:
(x + 1)(x + 1) = (x+1)^2 = x^2 + 2\cdot x \cdot 1 + 1^2 = x^2 + 2x + 1

(e) Difference of squares formula:
(x + y)(x - y) = x^2 - y^2