1. **Problem 1:** Simplify and factor $ (4x^2 - 16x + 7)(4x^2 - 16x + 15) + 16$. - Let $y = 4x^2 - 16x$. Rewrite expression as $(y + 7)(y + 15) + 16$ - Multiply: $y^2 + 22y + 105 + 16 = y^2 + 22y + 121$ - Recognize perfect square: $y^2 + 22y + 121 = (y + 11)^2$ - Substitute back: $(4x^2 - 16x + 11)^2$ 2. **Problem 2:** Simplify and factor $ (9x^2 + 9x - 4)(9x^2 + 9x - 10) - 72$. - Let $z = 9x^2 + 9x$. Expression becomes $(z - 4)(z - 10) - 72$ - Multiply: $z^2 - 14z + 40 - 72 = z^2 - 14z - 32$ - Factor quadratic: find factors of $-32$ summing to $-14$ are $-16$ and $2$ - So, $z^2 - 14z - 32 = (z - 16)(z + 2)$ - Substitute back: $(9x^2 + 9x - 16)(9x^2 + 9x + 2)$ 3. **Problem 3:** Simplify and factor $ (x + 2)(x + 4)(x + 6)(x + 8) - 9$. - Pair terms: $(x + 2)(x + 8) = x^2 + 10x + 16$ - $(x + 4)(x + 6) = x^2 + 10x + 24$ - Multiply: $(x^2 + 10x + 16)(x^2 + 10x + 24) - 9$ - Let $w = x^2 + 10x$, then: $(w + 16)(w + 24) - 9 = w^2 + 40w + 384 - 9 = w^2 + 40w + 375$ - Check discriminant $40^2 - 4 imes 375 = 1600 - 1500 = 100$, perfect square - Factor: $(w + 25)(w + 15)$ - Substitute back: $(x^2 + 10x + 25)(x^2 + 10x + 15)$ 4. **Problem 4:** Simplify and factor $ x(x + 1)(x + 2)(x + 3) + 1$. - Pair: $x(x+3) = x^2 + 3x$, $(x+1)(x+2) = x^2 + 3x + 2$ - Multiply: $(x^2 + 3x)(x^2 + 3x + 2) + 1$ - Let $u = x^2 + 3x$, expression is $u(u + 2) + 1 = u^2 + 2u + 1 = (u + 1)^2$ - Substitute back: $(x^2 + 3x + 1)^2$ 5. **Problem 5:** Simplify and factor $ (x + 1)(x + 2)(x + 3)(x + 6) - 3x^2$. - Pair: $(x + 1)(x + 6) = x^2 + 7x + 6$ - $(x + 2)(x + 3) = x^2 + 5x + 6$ - Multiply: $(x^2 + 7x + 6)(x^2 + 5x + 6) - 3x^2$ - Multiply polynomials: $$x^4 + 12x^3 + 53x^2 + 72x + 36 - 3x^2 = x^4 + 12x^3 + 50x^2 + 72x + 36$$ - Factor by grouping or trial (look for roots or factorization): - Use substitution or try possible rational roots: $x^4 + 12x^3 + 50x^2 + 72x + 36$ - Testing $x = -1$ gives: $$1 -12 + 50 - 72 + 36 = 3 eq 0$$ - Testing $x = -2$: $$16 - 96 + 200 - 144 + 36 = 12 eq 0$$ - No easy rational roots; use quadratic in form $(x^2 + ax + b)(x^2 + cx + d)$. - Find $a,c,b,d$ such that: - $a + c = 12$ - $ac + b + d = 50$ - $ad + bc = 72$ - $bd = 36$ - Try $b = 6$, $d = 6$ (since 36 factors as 6*6) and $a = 6$, $c = 6$: - Verify: - $6 + 6 = 12$ ✔ - $6 imes 6 + 6 + 6 = 36 + 12 = 48 eq 50$ - Try $b = 9$, $d = 4$: - Then $bd = 36$ ✔ - $a + c = 12$ - $ac + 9 + 4 = 50 ightarrow ac = 37$ - $ad + bc = ?$ with $a,c$ unknown yet - Try $a = 9$, $c=3$: - $a+c=12$ ✔ - $ac + 9 + 4 = 27 + 13 = 40 eq 50$ - Try $a=10$, $c=2$: - $a+c=12$ ✔ - $ac + 9 + 4 = 20 + 13 = 33 eq 50$ - Try $b=12$, $d=3$: - $bd=36$ ✔ - $ac + 12 + 3 = 50 ightarrow ac = 35$ - Try $a=7$, $c=5$: - $a+c=12$ ✔ - $ac + 12 +3 = 35 + 15 = 50$ ✔ - $ad + bc = 7 imes 3 + 5 imes 12 = 21 + 60 = 81 eq 72$ - Try swapped $b=3$, $d=12$: - $bd=36$ ✔ - $a+c=12$ - $ac + 3 + 12 = 50 ightarrow ac = 35$ - check $ad + bc = a imes 12 + b imes c = 12a + 3c = 72$ - Set $a=5$, $c=7$ then: - $a+c=12$ ✔ - $ac + 3 + 12 = 35 + 15 = 50$ ✔ - $12 imes 5 + 3 imes 7 = 60 + 21 = 81 eq 72$ - Try $a=6$, $c=6$: - $a+c=12$ ✔ - $ac + 3 + 12 = 36 + 15 = 51 eq 50$ - Try $b=18$, $d=2$: - $bd=36$ ✔ - $ac + 18 + 2 = 50 ightarrow ac = 30$ - test $a=10$, $c=2$: - $a+c=12$, $ac=20$ no - test $a=6$, $c=6$: - $a+c=12$, $ac=36$ - No exact neat factorization with integers, so the expression is already simplified as: $$x^4 + 12x^3 + 50x^2 + 72x + 36$$ 6. **Problem 6:** Factor $64x^3 - 144x^2y + 108xy^2 - 27y^3$. - Recognize as a cubic expression in $x$ and $y$. - Try to factor as a cube of a binomial: $$64x^3 = (4x)^3$$ $$27y^3 = (3y)^3$$ - Check if expression fits formula: $$a^3 - 3a^2b + 3ab^2 - b^3 = (a - b)^3$$ - Compare to given: $$64x^3 - 144x^2y + 108xy^2 - 27y^3$$ - Note: $$-144x^2y = -3 imes 64x^2 imes rac{3y}{4x} = -3 imes (4x)^2 imes (3y)$$ - Similarly: $$108xy^2 = 3 imes 4x imes (3y)^2$$ - So the expression is: $$(4x - 3y)^3$$ **Final Answers:** 1. $$(4x^2 - 16x + 11)^2$$ 2. $$(9x^2 + 9x - 16)(9x^2 + 9x + 2)$$ 3. $$(x^2 + 10x + 25)(x^2 + 10x + 15)$$ 4. $$(x^2 + 3x + 1)^2$$ 5. $$x^4 + 12x^3 + 50x^2 + 72x + 36$$ (cannot be factored nicely with integers) 6. $$(4x - 3y)^3$$