Question

1. $n^2 + 6n + 8$

$b^2 - 6b + 8$

1. $5n^2 + 10n + 20$

) $2n^2 + 6n - 108$

1. $a^2 - a - 90$
2. $2k^2 + 22k + 60$
3. $5v^2 - 30v + 40$
4. $p^2 + 11p + 10$
5. $4v^2 - 4v - 8$
6. $2p^2 + 2p - 4$
7. $v^2 - 7v + 10$
8. $x^2 - 15x + 50$
9. $6v^2 + 66v + 60$
10. $p^2 + 3p - 18$

Explanation:

All problems involve factoring quadratic expressions. Below is the step-by-step factoring for each:

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Problem: $b^2 - 6b + 8$

Step1: Find pair for sum -6, product 8

Numbers: $-2$ and $-4$

Step2: Write factored form

$(b-2)(b-4)$

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Problem 14: $n^2 + 6n + 8$

Step1: Find pair for sum 6, product 8

Numbers: $2$ and $4$

Step2: Write factored form

$(n+2)(n+4)$

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Problem: $2n^2 + 6n - 108$

Step1: Factor out GCF 2

$2(n^2 + 3n - 54)$

Step2: Factor quadratic inside

Find pair for sum 3, product -54: $9$ and $-6$

Step3: Write final factored form

$2(n+9)(n-6)$

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Problem 16: $5n^2 + 10n + 20$

Step1: Factor out GCF 5

$5(n^2 + 2n + 4)$

Step2: Check quadratic (no real factors)

Quadratic is irreducible over reals

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Problem 17: $2k^2 + 22k + 60$

Step1: Factor out GCF 2

$2(k^2 + 11k + 30)$

Step2: Factor quadratic inside

Find pair for sum 11, product 30: $5$ and $6$

Step3: Write final factored form

$2(k+5)(k+6)$

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Problem 18: $a^2 - a - 90$

Step1: Find pair for sum -1, product -90

Numbers: $9$ and $-10$

Step2: Write factored form

$(a+9)(a-10)$

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Problem 19: $p^2 + 11p + 10$

Step1: Find pair for sum 11, product 10

Numbers: $1$ and $10$

Step2: Write factored form

$(p+1)(p+10)$

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Problem 20: $5v^2 - 30v + 40$

Step1: Factor out GCF 5

$5(v^2 - 6v + 8)$

Step2: Factor quadratic inside

Find pair for sum -6, product 8: $-2$ and $-4$

Step3: Write final factored form

$5(v-2)(v-4)$

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Problem 21: $2p^2 + 2p - 4$

Step1: Factor out GCF 2

$2(p^2 + p - 2)$

Step2: Factor quadratic inside

Find pair for sum 1, product -2: $2$ and $-1$

Step3: Write final factored form

$2(p+2)(p-1)$

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Problem 22: $4v^2 - 4v - 8$

Step1: Factor out GCF 4

$4(v^2 - v - 2)$

Step2: Factor quadratic inside

Find pair for sum -1, product -2: $1$ and $-2$

Step3: Write final factored form

$4(v+1)(v-2)$

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Problem 23: $x^2 - 15x + 50$

Step1: Find pair for sum -15, product 50

Numbers: $-5$ and $-10$

Step2: Write factored form

$(x-5)(x-10)$

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Problem 24: $v^2 - 7v + 10$

Step1: Find pair for sum -7, product 10

Numbers: $-2$ and $-5$

Step2: Write factored form

$(v-2)(v-5)$

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Problem 25: $p^2 + 3p - 18$

Step1: Find pair for sum 3, product -18

Numbers: $6$ and $-3$

Step2: Write factored form

$(p+6)(p-3)$

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Problem 26: $6v^2 + 66v + 60$

Step1: Factor out GCF 6

$6(v^2 + 11v + 10)$

Step2: Factor quadratic inside

Find pair for sum 11, product 10: $1$ and $10$

Step3: Write final factored form

$6(v+1)(v+10)$

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Answer:

1. $b^2 - 6b + 8 = (b-2)(b-4)$
2. $n^2 + 6n + 8 = (n+2)(n+4)$
3. $2n^2 + 6n - 108 = 2(n+9)(n-6)$
4. $5n^2 + 10n + 20 = 5(n^2 + 2n + 4)$
5. $2k^2 + 22k + 60 = 2(k+5)(k+6)$
6. $a^2 - a - 90 = (a+9)(a-10)$
7. $p^2 + 11p + 10 = (p+1)(p+10)$
8. $5v^2 - 30v + 40 = 5(v-2)(v-4)$
9. $2p^2 + 2p - 4 = 2(p+2)(p-1)$
10. $4v^2 - 4v - 8 = 4(v+1)(v-2)$
11. $x^2 - 15x + 50 = (x-5)(x-10)$
12. $v^2 - 7v + 10 = (v-2)(v-5)$
13. $p^2 + 3p - 18 = (p+6)(p-3)$
14. $6v^2 + 66v + 60 = 6(v+1)(v+10)$