1. **Problem:** Expand and simplify the expression $(8r - 3)(3r + 8)$. **Formula:** Use distributive property (FOIL): $(a+b)(c+d) = ac + ad + bc + bd$. **Work:** $$8r \times 3r = 24r^2$$ $$8r \times 8 = 64r$$ $$-3 \times 3r = -9r$$ $$-3 \times 8 = -24$$ Sum: $$24r^2 + 64r - 9r - 24 = 24r^2 + 55r - 24$$ 2. **Problem:** Expand and simplify $(4n - 5)(-7n + 5)$. **Work:** $$4n \times (-7n) = -28n^2$$ $$4n \times 5 = 20n$$ $$-5 \times (-7n) = 35n$$ $$-5 \times 5 = -25$$ Sum: $$-28n^2 + 20n + 35n - 25 = -28n^2 + 55n - 25$$ 3. **Problem:** Expand and simplify $(4p^2 + p + 7)(p + 4)$. **Work:** $$4p^2 \times p = 4p^3$$ $$4p^2 \times 4 = 16p^2$$ $$p \times p = p^2$$ $$p \times 4 = 4p$$ $$7 \times p = 7p$$ $$7 \times 4 = 28$$ Sum: $$4p^3 + 16p^2 + p^2 + 4p + 7p + 28 = 4p^3 + 17p^2 + 11p + 28$$ 4. **Problem:** Expand and simplify $(5n^2 - 3n + 3)(4n^2 + 6n)$. **Work:** $$5n^2 \times 4n^2 = 20n^4$$ $$5n^2 \times 6n = 30n^3$$ $$-3n \times 4n^2 = -12n^3$$ $$-3n \times 6n = -18n^2$$ $$3 \times 4n^2 = 12n^2$$ $$3 \times 6n = 18n$$ Sum: $$20n^4 + 30n^3 - 12n^3 - 18n^2 + 12n^2 + 18n = 20n^4 + 18n^3 - 6n^2 + 18n$$ 5. **Problem:** Expand and simplify $(2i^3 - 2i + 5)(i^2 - 7)$. **Work:** $$2i^3 \times i^2 = 2i^5$$ $$2i^3 \times (-7) = -14i^3$$ $$-2i \times i^2 = -2i^3$$ $$-2i \times (-7) = 14i$$ $$5 \times i^2 = 5i^2$$ $$5 \times (-7) = -35$$ Sum: $$2i^5 - 14i^3 - 2i^3 + 14i + 5i^2 - 35 = 2i^5 - 16i^3 + 5i^2 + 14i - 35$$ 6. **Problem:** Expand and simplify $(3e^4 - 3e + 4)(e^3 - 6)$. **Work:** $$3e^4 \times e^3 = 3e^7$$ $$3e^4 \times (-6) = -18e^4$$ $$-3e \times e^3 = -3e^4$$ $$-3e \times (-6) = 18e$$ $$4 \times e^3 = 4e^3$$ $$4 \times (-6) = -24$$ Sum: $$3e^7 - 18e^4 - 3e^4 + 18e + 4e^3 - 24 = 3e^7 - 21e^4 + 4e^3 + 18e - 24$$ 7. **Problem:** Expand and simplify $(-2n + 4)(6n + 0.7)$. **Work:** $$-2n \times 6n = -12n^2$$ $$-2n \times 0.7 = -1.4n$$ $$4 \times 6n = 24n$$ $$4 \times 0.7 = 2.8$$ Sum: $$-12n^2 - 1.4n + 24n + 2.8 = -12n^2 + 22.6n + 2.8$$ 8. **Problem:** Expand and simplify $(3i^3 + 4)(i^3 - u)$. **Work:** $$3i^3 \times i^3 = 3i^6$$ $$3i^3 \times (-u) = -3ui^3$$ $$4 \times i^3 = 4i^3$$ $$4 \times (-u) = -4u$$ Sum: $$3i^6 - 3ui^3 + 4i^3 - 4u = 3i^6 + (4 - 3u)i^3 - 4u$$ 9. **Problem:** Expand and simplify $(-2fk - k)(-2fk - k)$. **Work:** This is $(a+b)^2$ with $a = -2fk$, $b = -k$. $$(-2fk)^2 = 4f^2k^2$$ $$2 \times (-2fk) \times (-k) = 4fk^2$$ $$(-k)^2 = k^2$$ Sum: $$4f^2k^2 + 4fk^2 + k^2 = k^2(4f^2 + 4f + 1)$$ 10. **Problem:** Expand and simplify $\left(\frac{1}{3} t^3 - 3\right)(5t - 5)$. **Work:** $$\frac{1}{3} t^3 \times 5t = \frac{5}{3} t^4$$ $$\frac{1}{3} t^3 \times (-5) = -\frac{5}{3} t^3$$ $$-3 \times 5t = -15t$$ $$-3 \times (-5) = 15$$ Sum: $$\frac{5}{3} t^4 - \frac{5}{3} t^3 - 15t + 15$$