spiral 3 lesson 5 - simple trinomial factoring - special cases
factor each completely.
1) $6x^{2}+12x - 378$ 2) $2b^{2}-38b + 180$

3) $2x^{2}-14x + 12$ 4) $2a^{2}-14a + 20$

5) $6x^{2}-42x - 48$ 6) $-6a^{2}-30a - 24$

7) $-6n^{2}-36n$ 8) $2r^{2}-32r + 120$

9) $5n^{2}-15n - 140$ 10) $-4x^{2}-8x$

Question

spiral 3 lesson 5 - simple trinomial factoring - special cases
factor each completely.
1) $6x^{2}+12x - 378$  					2) $2b^{2}-38b + 180$

3) $2x^{2}-14x + 12$  					4) $2a^{2}-14a + 20$

5) $6x^{2}-42x - 48$  					6) $-6a^{2}-30a - 24$

7) $-6n^{2}-36n$  						8) $2r^{2}-32r + 120$

9) $5n^{2}-15n - 140$  					10) $-4x^{2}-8x$

Accepted Answer

Let's solve each problem one by one using factoring techniques (greatest common factor, then factoring the quadratic):

### 1) $ 6x^2 + 12x - 378 $
## Step 1: Factor out GCF (6)
$ 6(x^2 + 2x - 63) $
## Step 2: Factor the quadratic $ x^2 + 2x - 63 $
Find two numbers that multiply to -63 and add to 2: 9 and -7.
$ x^2 + 2x - 63 = (x + 9)(x - 7) $
## Step 3: Combine
$ 6(x + 9)(x - 7) $
# Answer: $ 6(x + 9)(x - 7) $

### 2) $ 2b^2 - 38b + 180 $
## Step 1: Factor out GCF (2)
$ 2(b^2 - 19b + 90) $
## Step 2: Factor the quadratic $ b^2 - 19b + 90 $
Find two numbers that multiply to 90 and add to -19: -9 and -10.
$ b^2 - 19b + 90 = (b - 9)(b - 10) $
## Step 3: Combine
$ 2(b - 9)(b - 10) $
# Answer: $ 2(b - 9)(b - 10) $

### 3) $ 2x^2 - 14x + 12 $
## Step 1: Factor out GCF (2)
$ 2(x^2 - 7x + 6) $
## Step 2: Factor the quadratic $ x^2 - 7x + 6 $
Find two numbers that multiply to 6 and add to -7: -1 and -6.
$ x^2 - 7x + 6 = (x - 1)(x - 6) $
## Step 3: Combine
$ 2(x - 1)(x - 6) $
# Answer: $ 2(x - 1)(x - 6) $

### 4) $ 2a^2 - 14a + 20 $
## Step 1: Factor out GCF (2)
$ 2(a^2 - 7a + 10) $
## Step 2: Factor the quadratic $ a^2 - 7a + 10 $
Find two numbers that multiply to 10 and add to -7: -2 and -5.
$ a^2 - 7a + 10 = (a - 2)(a - 5) $
## Step 3: Combine
$ 2(a - 2)(a - 5) $
# Answer: $ 2(a - 2)(a - 5) $

### 5) $ 6x^2 - 42x - 48 $
## Step 1: Factor out GCF (6)
$ 6(x^2 - 7x - 8) $
## Step 2: Factor the quadratic $ x^2 - 7x - 8 $
Find two numbers that multiply to -8 and add to -7: -8 and 1.
$ x^2 - 7x - 8 = (x - 8)(x + 1) $
## Step 3: Combine
$ 6(x - 8)(x + 1) $
# Answer: $ 6(x - 8)(x + 1) $

### 6) $ -6a^2 - 30a - 24 $
## Step 1: Factor out GCF (-6)
$ -6(a^2 + 5a + 4) $
## Step 2: Factor the quadratic $ a^2 + 5a + 4 $
Find two numbers that multiply to 4 and add to 5: 1 and 4.
$ a^2 + 5a + 4 = (a + 1)(a + 4) $
## Step 3: Combine
$ -6(a + 1)(a + 4) $
# Answer: $ -6(a + 1)(a + 4) $

### 7) $ -6n^2 - 36n $
## Step 1: Factor out GCF (-6n)
$ -6n(n + 6) $
# Answer: $ -6n(n + 6) $

### 8) $ 2r^2 - 32r + 120 $
## Step 1: Factor out GCF (2)
$ 2(r^2 - 16r + 60) $
## Step 2: Factor the quadratic $ r^2 - 16r + 60 $
Find two numbers that multiply to 60 and add to -16: -6 and -10.
$ r^2 - 16r + 60 = (r - 6)(r - 10) $
## Step 3: Combine
$ 2(r - 6)(r - 10) $
# Answer: $ 2(r - 6)(r - 10) $

### 9) $ 5n^2 - 15n - 140 $
## Step 1: Factor out GCF (5)
$ 5(n^2 - 3n - 28) $
## Step 2: Factor the quadratic $ n^2 - 3n - 28 $
Find two numbers that multiply to -28 and add to -3: -7 and 4.
$ n^2 - 3n - 28 = (n - 7)(n + 4) $
## Step 3: Combine
$ 5(n - 7)(n + 4) $
# Answer: $ 5(n - 7)(n + 4) $

### 10) $ -4x^2 - 8x $
## Step 1: Factor out GCF (-4x)
$ -4x(x + 2) $
# Answer: $ -4x(x + 2) $

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QUESTION IMAGE

Question

Explanation:

1) \( 6x^2 + 12x - 378 \)

Step 1: Factor out GCF (6)

Step 2: Factor the quadratic \( x^2 + 2x - 63 \)

Step 3: Combine

Answer:

2) \( 2b^2 - 38b + 180 \)

Step 1: Factor out GCF (2)

Step 2: Factor the quadratic \( b^2 - 19b + 90 \)

Step 3: Combine