Question

what is $(8b + 3)(b - 1)?$
○ a. $8b^{2}-5b - 3$
○ b. $8b^{2}-5b - 2$
○ c. $8b^{2}-6b - 3$
○ d. $8b^{2}-6b - 4$

what is $(2n - 1)(7n^{2}-n - 6)?$
○ a. $14n^{3}-9n^{2}-11n + 6$
○ b. $14n^{3}-8n^{2}-10n + 6$
○ c. $14n^{3}-9n^{2}-11n + 5$
○ d. $14n^{3}-9n^{2}-10n + 6$

what is $(2a^{2}-3a + 1)(5a^{2}+a - 2)?$
○ a. $10a^{4}-13a^{3}-3a^{2}+7a - 2$
○ b. $10a^{4}-13a^{3}-2a^{2}+6a - 2$
○ c. $10a^{4}-13a^{3}-2a^{2}+7a - 2$
○ d. $10a^{4}-12a^{3}-1a^{2}+7a - 2$

Explanation:

Step1: Expand first binomial product

Use distributive property (FOIL):
$$(8b+3)(b-1) = 8b\cdot b + 8b\cdot(-1) + 3\cdot b + 3\cdot(-1)$$
$$=8b^2 -8b +3b -3$$
$$=8b^2 -5b -3$$

Step2: Expand second binomial product

Use distributive property:
$$(2n-1)(7n^2 -n -6) = 2n(7n^2 -n -6) -1(7n^2 -n -6)$$
$$=14n^3 -2n^2 -12n -7n^2 +n +6$$
$$=14n^3 -9n^2 -11n +6$$

Step3: Expand third polynomial product

Use distributive property:
$$(2a^2 -3a +1)(5a^2 +a -2) = 2a^2(5a^2 +a -2) -3a(5a^2 +a -2) +1(5a^2 +a -2)$$
$$=10a^4 +2a^3 -4a^2 -15a^3 -3a^2 +6a +5a^2 +a -2$$
$$=10a^4 -13a^3 -2a^2 +7a -2$$

Answer:

1. a. $8b^2 -5b -3$
2. a. $14n^3 -9n^2 -11n +6$
3. c. $10a^4 -13a^3 -2a^2 +7a -2$