Question

procedure:

1. find the factors of the constant, c
2. find the factors of c whose sum is b
3. rewrite the polynomial as factors

1. \\( x^2 + 9x + 20 \\) \\( x^2 + 8x + 12 \\)
2. \\( x^2 + 13x + 42 \\) \\( x^2 + 10x + 16 \\)
3. \\( x^2 + 5x + 6 \\) \\( x^2 + 7x + 6 \\)
4. \\( x^2 + 11x + 10 \\) \\( x^2 + 7x + 10 \\)
5. \\( x^2 + 6x + 8 \\) \\( x^2 + 2x + 1 \\)
6. \\( x^2 + 7x + 12 \\) \\( x^2 + 15x + 54 \\)

Explanation:

Step1: Factor $x^2+8x+12$

Find factors of 12 that sum to 8: 2 and 6.
$\boldsymbol{x^2+8x+12=(x+2)(x+6)}$

Step2: Factor $x^2+10x+16$

Find factors of 16 that sum to 10: 2 and 8.
$\boldsymbol{x^2+10x+16=(x+2)(x+8)}$

Step3: Factor $x^2+7x+6$

Find factors of 6 that sum to 7: 1 and 6.
$\boldsymbol{x^2+7x+6=(x+1)(x+6)}$

Step4: Factor $x^2+7x+10$

Find factors of 10 that sum to 7: 2 and 5.
$\boldsymbol{x^2+7x+10=(x+2)(x+5)}$

Step5: Factor $x^2+2x+1$

Find factors of 1 that sum to 2: 1 and 1.
$\boldsymbol{x^2+2x+1=(x+1)^2}$

Step6: Factor $x^2+15x+54$

Find factors of 54 that sum to 15: 6 and 9.
$\boldsymbol{x^2+15x+54=(x+6)(x+9)}$

Answer:

1. $x^2+9x+20=(x+4)(x+5)$; $x^2+8x+12=(x+2)(x+6)$
2. $x^2+13x+42=(x+6)(x+7)$; $x^2+10x+16=(x+2)(x+8)$
3. $x^2+5x+6=(x+2)(x+3)$; $x^2+7x+6=(x+1)(x+6)$
4. $x^2+11x+10=(x+1)(x+10)$; $x^2+7x+10=(x+2)(x+5)$
5. $x^2+6x+8=(x+2)(x+4)$; $x^2+2x+1=(x+1)^2$
6. $x^2+7x+12=(x+3)(x+4)$; $x^2+15x+54=(x+6)(x+9)$