Question

factoring mixed practice #1
factor:
① $x^{2}+6x + 8$ ② $x^{2}+6x - 16$ ③ $x^{2}-6x - 16$
④ $x^{2}-25$ ⑤ $4x^{3}+8x^{2}$ ⑥ $3x^{2}+31x + 36$
⑦ $x^{2}+9x + 20$ ⑧ $x^{2}-9x + 20$ ⑨ $x^{2}-x - 6$
⑩ $4x + 10$ ⑪ $2x^{2}+x - 3$ ⑫ $x^{2}+24x + 144$
⑬ $4x^{2}-9$ ⑭ $3x^{2}+15x + 18$ ⑮ $4x^{2}-4x + 1$

Explanation:

Step1: Factor $x^2+6x+8$

Find two numbers: $2$ and $4$ (sum=6, product=8).
$\boldsymbol{x^2+6x+8=(x+2)(x+4)}$

Step2: Factor $x^2+6x-16$

Find two numbers: $8$ and $-2$ (sum=6, product=-16).
$\boldsymbol{x^2+6x-16=(x+8)(x-2)}$

Step3: Factor $x^2-6x-16$

Find two numbers: $-8$ and $2$ (sum=-6, product=-16).
$\boldsymbol{x^2-6x-16=(x-8)(x+2)}$

Step4: Factor $x^2-25$

Use difference of squares: $a^2-b^2=(a-b)(a+b)$.
$\boldsymbol{x^2-25=(x-5)(x+5)}$

Step5: Factor $4x^3+8x^2$

Factor out GCF $4x^2$.
$\boldsymbol{4x^3+8x^2=4x^2(x+2)}$

Step6: Factor $3x^2+31x+36$

Split middle term: $31x=27x+4x$, factor by grouping.
$3x^2+27x+4x+36=3x(x+9)+4(x+9)$
$\boldsymbol{3x^2+31x+36=(3x+4)(x+9)}$

Step7: Factor $x^2+9x+20$

Find two numbers: $4$ and $5$ (sum=9, product=20).
$\boldsymbol{x^2+9x+20=(x+4)(x+5)}$

Step8: Factor $x^2-9x+20$

Find two numbers: $-4$ and $-5$ (sum=-9, product=20).
$\boldsymbol{x^2-9x+20=(x-4)(x-5)}$

Step9: Factor $x^2-x-6$

Find two numbers: $-3$ and $2$ (sum=-1, product=-6).
$\boldsymbol{x^2-x-6=(x-3)(x+2)}$

Step10: Factor $4x+10$

Factor out GCF $2$.
$\boldsymbol{4x+10=2(2x+5)}$

Step11: Factor $2x^2+x-3$

Split middle term: $x=3x-2x$, factor by grouping.
$2x^2+3x-2x-3=x(2x+3)-1(2x+3)$
$\boldsymbol{2x^2+x-3=(x-1)(2x+3)}$

Step12: Factor $x^2+24x+144$

Perfect square trinomial: $(a+b)^2=a^2+2ab+b^2$.
$\boldsymbol{x^2+24x+144=(x+12)^2}$

Step13: Factor $4x^2-9$

Difference of squares: $a^2-b^2=(a-b)(a+b)$.
$\boldsymbol{4x^2-9=(2x-3)(2x+3)}$

Step14: Factor $3x^2+15x+18$

Factor out GCF $3$, then factor quadratic.
$3(x^2+5x+6)=3(x+2)(x+3)$
$\boldsymbol{3x^2+15x+18=3(x+2)(x+3)}$

Step15: Factor $4x^2-4x+1$

Perfect square trinomial: $(a-b)^2=a^2-2ab+b^2$.
$\boldsymbol{4x^2-4x+1=(2x-1)^2}$

Answer:

1. $\boldsymbol{(x+2)(x+4)}$
2. $\boldsymbol{(x+8)(x-2)}$
3. $\boldsymbol{(x-8)(x+2)}$
4. $\boldsymbol{(x-5)(x+5)}$
5. $\boldsymbol{4x^2(x+2)}$
6. $\boldsymbol{(3x+4)(x+9)}$
7. $\boldsymbol{(x+4)(x+5)}$
8. $\boldsymbol{(x-4)(x-5)}$
9. $\boldsymbol{(x-3)(x+2)}$
10. $\boldsymbol{2(2x+5)}$
11. $\boldsymbol{(x-1)(2x+3)}$
12. $\boldsymbol{(x+12)^2}$
13. $\boldsymbol{(2x-3)(2x+3)}$
14. $\boldsymbol{3(x+2)(x+3)}$
15. $\boldsymbol{(2x-1)^2}$