Sure! Let's factor the given quadratic expression step-by-step. Here is the given expression:
[tex]\[ 6x^2 - 3x - 30 \][/tex]
### Step 1: Factor out the Greatest Common Factor (GCF)
The first step in factoring is to identify and factor out the greatest common factor (GCF) from all terms. In this expression, each term can be divided by 3:
[tex]\[ 6x^2 - 3x - 30 = 3(2x^2 - x - 10) \][/tex]
### Step 2: Factor the quadratic expression inside the parentheses
Now we need to factor the quadratic expression [tex]\(2x^2 - x - 10\)[/tex].
#### Step 2.1: Identify coefficients
For [tex]\(2x^2 - x - 10\)[/tex], the coefficients are:
- [tex]\(a = 2\)[/tex] (coefficient of [tex]\(x^2\)[/tex])
- [tex]\(b = -1\)[/tex] (coefficient of [tex]\(x\)[/tex])
- [tex]\(c = -10\)[/tex] (constant term)
#### Step 2.2: Find two numbers that multiply to [tex]\(ac\)[/tex] and add to [tex]\(b\)[/tex]
We need to find two numbers that multiply to [tex]\(a \cdot c = 2 \cdot (-10) = -20\)[/tex] and add to [tex]\(b = -1\)[/tex]. These two numbers are 4 and -5, since:
[tex]\[ 4 \cdot (-5) = -20 \][/tex]
[tex]\[ 4 + (-5) = -1 \][/tex]
#### Step 2.3: Split the middle term using these numbers
We can rewrite [tex]\(-x\)[/tex] as [tex]\(4x - 5x\)[/tex]:
[tex]\[ 2x^2 - x - 10 = 2x^2 + 4x - 5x - 10 \][/tex]
#### Step 2.4: Factor by grouping
Now we group the terms and factor each group:
[tex]\[ = (2x^2 + 4x) + (-5x - 10) \][/tex]
[tex]\[ = 2x(x + 2) - 5(x + 2) \][/tex]
#### Step 2.5: Factor out the common binomial factor
Both terms contain the binomial factor [tex]\((x + 2)\)[/tex]:
[tex]\[ = (2x - 5)(x + 2) \][/tex]
### Step 3: Combine the factored expressions
We previously factored out a 3, so now we combine this with our newly factored quadratic:
[tex]\[ 6x^2 - 3x - 30 = 3(2x^2 - x - 10) = 3(2x - 5)(x + 2) \][/tex]
### Final Answer:
The completely factored form of the given expression is:
[tex]\[ 3(2x - 5)(x + 2) \][/tex]
So, to place the answer in the proper location on the grid, you should enter:
[tex]\[ 3(2x - 5)(x + 2) \][/tex]
