Question

1. $x^{2}-x-20$
2. $9x^{2}-1$
3. $10x^{2}+15x$
4. $4x^{3}+12x^{2}+8x$
5. $2x^{2}+14x+20$
6. $-2x^{3}+18x$
7. $16x^{4}-1$
8. $x^{2}-10x+21$
9. $x^{2}+5x-50$

Explanation:

10) Step1: Find two factors of -20

Find two numbers that multiply to $-20$ and add to $-1$: $-5$ and $4$.

10) Step2: Factor the quadratic

$x^2 - x - 20 = (x - 5)(x + 4)$

11) Step1: Recognize difference of squares

$9x^2 - 1 = (3x)^2 - (1)^2$

11) Step2: Apply difference of squares rule

$a^2 - b^2 = (a-b)(a+b)$, so $(3x - 1)(3x + 1)$

12) Step1: Factor out greatest common factor

GCF of $10x^2$ and $15x$ is $5x$.
$10x^2 + 15x = 5x(2x + 3)$

13) Step1: Factor out GCF first

GCF of $4x^3, 12x^2, 8x$ is $4x$.
$4x^3 + 12x^2 + 8x = 4x(x^2 + 3x + 2)$

13) Step2: Factor the quadratic

Find factors of 2 that add to 3: 1 and 2.
$4x(x + 1)(x + 2)$

14) Step1: Factor out GCF

GCF of $2x^2, 14x, 20$ is $2$.
$2x^2 + 14x + 20 = 2(x^2 + 7x + 10)$

14) Step2: Factor the quadratic

Find factors of 10 that add to 7: 2 and 5.
$2(x + 2)(x + 5)$

15) Step1: Factor out GCF

GCF of $-2x^3$ and $18x$ is $-2x$.
$-2x^3 + 18x = -2x(x^2 - 9)$

15) Step2: Factor difference of squares

$x^2 - 9 = x^2 - 3^2$, so $-2x(x - 3)(x + 3)$

16) Step1: Recognize difference of squares

$16x^4 - 1 = (4x^2)^2 - (1)^2$

16) Step2: Apply difference of squares rule

$(4x^2 - 1)(4x^2 + 1)$

16) Step3: Factor remaining square

$4x^2 -1 = (2x)^2 -1^2$, so $(2x - 1)(2x + 1)(4x^2 + 1)$

17) Step1: Find factors of 21

Find two numbers that multiply to 21 and add to $-10$: $-3$ and $-7$.

17) Step2: Factor the quadratic

$x^2 - 10x + 21 = (x - 3)(x - 7)$

18) Step1: Find factors of -50

Find two numbers that multiply to $-50$ and add to 5: $10$ and $-5$.

18) Step2: Factor the quadratic

$x^2 + 5x - 50 = (x + 10)(x - 5)$

Answer:

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