Question

1. $8n^4 + 48n^3$
2. $45n^3 + 45n^2 - 350n$
3. $3r^3 + 2r^2 - 5r$
4. $3n^2 + 14n - 80$
5. $3n^4 + n^3 - 24n^2$
6. $4x^3 + 26x^2 + 42x$
7. $7m^2 - 26m - 45$
8. $7n^2 - 67n - 30$
9. $5x^2 - 19x + 12$
10. $7x^2 + 8x + 1$
11. $5m^2 - 27m - 56$
12. $10m^3 + 34m^2 + 28m$
13. $36x^2 + 96x$
14. $9x^2 + 14x - 8$
15. $16n^2 + 28n + 6$
16. $4k^2 - 27k + 18$
17. $40x^4 - 188x^3 + 168x^2$
18. $4n^2 + 49n + 90$
19. $8x^4 - 38x^3 + 24x^2$
20. $10x^2 + 3x - 1$
21. $4b^4 + 12b^3$
22. $9r^3 + 21r^2$

Explanation:

Step1: Factor out GCF of $8n^4+48n^3$

$\gcd(8,48)=8$, $\gcd(n^4,n^3)=n^3$
$8n^3(n + 6)$

Step2: Factor out GCF of $45n^3+45n^2-350n$

$\gcd(45,45,-350)=5$, $\gcd(n^3,n^2,n)=n$
$5n(9n^2 + 9n - 70)$
Factor quadratic: $9n^2+9n-70=(3n+14)(3n-5)$
$5n(3n+14)(3n-5)$

Step3: Factor out GCF of $3r^3+2r^2-5r$

$\gcd(3,2,-5)=1$, $\gcd(r^3,r^2,r)=r$
$r(3r^2 + 2r - 5)$
Factor quadratic: $3r^2+2r-5=(3r+5)(r-1)$
$r(3r+5)(r-1)$

Step4: Factor quadratic $3n^2+14n-80$

Find pairs: $20,-12$; split middle term
$3n^2+20n-12n-80=n(3n+20)-4(3n+20)$
$(3n+20)(n-4)$

Step5: Factor out GCF of $3n^4+n^3-24n^2$

$\gcd(3,1,-24)=1$, $\gcd(n^4,n^3,n^2)=n^2$
$n^2(3n^2 + n - 24)$
Factor quadratic: $3n^2+n-24=(3n-8)(n+3)$
$n^2(3n-8)(n+3)$

Step6: Factor out GCF of $4x^3+26x^2+42x$

$\gcd(4,26,42)=2$, $\gcd(x^3,x^2,x)=x$
$2x(2x^2 + 13x + 21)$
Factor quadratic: $2x^2+13x+21=(2x+7)(x+3)$
$2x(2x+7)(x+3)$

Step7: Factor quadratic $7m^2-26m-45$

Find pairs: $9,-35$; split middle term
$7m^2+9m-35m-45=m(7m+9)-5(7m+9)$
$(7m+9)(m-5)$

Step8: Factor quadratic $7n^2-67n-30$

Find pairs: $3,-70$; split middle term
$7n^2+3n-70n-30=n(7n+3)-10(7n+3)$
$(7n+3)(n-10)$

Step9: Factor quadratic $5x^2-19x+12$

Find pairs: $-4,-15$; split middle term
$5x^2-4x-15x+12=x(5x-4)-3(5x-4)$
$(5x-4)(x-3)$

Step10: Factor quadratic $7x^2+8x+1$

Find pairs: $7,1$; split middle term
$7x^2+7x+x+1=7x(x+1)+1(x+1)$
$(7x+1)(x+1)$

Step11: Factor quadratic $5m^2-27m-56$

Find pairs: $8,-35$; split middle term
$5m^2+8m-35m-56=m(5m+8)-7(5m+8)$
$(5m+8)(m-7)$

Step12: Factor out GCF of $10m^3+34m^2+28m$

$\gcd(10,34,28)=2$, $\gcd(m^3,m^2,m)=m$
$2m(5m^2 + 17m + 14)$
Factor quadratic: $5m^2+17m+14=(5m+7)(m+2)$
$2m(5m+7)(m+2)$

Step13: Factor out GCF of $36x^2+96x$

$\gcd(36,96)=12$, $\gcd(x^2,x)=x$
$12x(3x + 8)$

Step14: Factor quadratic $9x^2+14x-8$

Find pairs: $18,-4$; split middle term
$9x^2+18x-4x-8=9x(x+2)-4(x+2)$
$(9x-4)(x+2)$

Step15: Factor out GCF of $16n^2+28n+6$

$\gcd(16,28,6)=2$
$2(8n^2 + 14n + 3)$
Factor quadratic: $8n^2+14n+3=(4n+1)(2n+3)$
$2(4n+1)(2n+3)$

Step16: Factor quadratic $4k^2-27k+18$

Find pairs: $-3,-24$; split middle term
$4k^2-3k-24k+18=k(4k-3)-6(4k-3)$
$(4k-3)(k-6)$

Step17: Factor out GCF of $40x^4-188x^3+168x^2$

$\gcd(40,188,168)=4$, $\gcd(x^4,x^3,x^2)=x^2$
$4x^2(10x^2 - 47x + 42)$
Factor quadratic: $10x^2-47x+42=(5x-6)(2x-7)$
$4x^2(5x-6)(2x-7)$

Step18: Factor quadratic $4n^2+49n+90$

Find pairs: $40,9$; split middle term
$4n^2+40n+9n+90=4n(n+10)+9(n+10)$
$(4n+9)(n+10)$

Step19: Factor out GCF of $8x^4-38x^3+24x^2$

$\gcd(8,38,24)=2$, $\gcd(x^4,x^3,x^2)=x^2$
$2x^2(4x^2 - 19x + 12)$
Factor quadratic: $4x^2-19x+12=(4x-3)(x-4)$
$2x^2(4x-3)(x-4)$

Step20: Factor quadratic $10x^2+3x-1$

Find pairs: $5,-2$; split middle term
$10x^2+5x-2x-1=5x(2x+1)-1(2x+1)$
$(5x-1)(2x+1)$

Step21: Factor out GCF of $4b^4+12b^3$

$\gcd(4,12)=4$, $\gcd(b^4,b^3)=b^3$
$4b^3(b + 3)$

Step22: Factor out GCF of $9r^3+21r^2$

$\gcd(9,21)=3$, $\gcd(r^3,r^2)=r^2$
$3r^2(3r + 7)$

Answer:

1. $8n^3(n + 6)$
2. $5n(3n+14)(3n-5)$
3. $r(3r+5)(r-1)$
4. $(3n+20)(n-4)$
5. $n^2(3n-8)(n+3)$
6. $2x(2x+7)(x+3)$
7. $(7m+9)(m-5)$
8. $(7n+3)(n-10)$
9. $(5x-4)(x-3)$
10. $(7x+1)(x+1)$
11. $(5m+8)(m-7)$
12. $2m(5m+7)(m+2)$
13. $12x(3x + 8)$
14. $(9x-4)(x+2)$
15. $2(4n+1)(2n+3)$
16. $(4k-3)(k-6)$
17. $4x^2(5x-6)(2x-7)$
18. $(4n+9)(n+10)$
19. $2x^2(4x-3)(x-4)$
20. $(5x-1)(2x+1)$
21. $4b^3(b + 3)$
22. $3r^2(3r + 7)$