Question

factor each completely.
3n² − 15n + 18
3(n + 9)(n + 1)
6(n - 10)(n + 2)
3(n + 5)(n + 2)
3(n - 3)(n - 2)
3(n + 1)(n + 2)

Explanation:

Step1: Factor out GCF

The given expression is \(3n^2 - 15n + 18\). First, factor out the greatest common factor (GCF), which is 3. So we get \(3(n^2 - 5n + 6)\).

Step2: Factor the quadratic

Now, factor the quadratic \(n^2 - 5n + 6\). We need two numbers that multiply to 6 and add up to -5. Those numbers are -3 and -2. So, \(n^2 - 5n + 6=(n - 3)(n - 2)\).

Step3: Combine the factors

Putting it back together with the GCF we factored out earlier, we have \(3(n - 3)(n - 2)\). Wait, but let's check the options. Wait, maybe I made a mistake. Wait, let's expand the options to check. Let's check \(3(n - 3)(n - 2)\) vs the options. Wait, the option \(3(n - 3)(n - 2)\) is one of the options? Wait, the options are:

1. \(5(n + 9)(n + 1)\) – no.
2. \(6(n - 10)(n + 2)\) – no.
3. \(3(n + 5)(n + 2)\) – no.
4. \(3(n - 3)(n - 2)\) – wait, but in the image, the fourth option is \(3(n - 3)(n - 2)\)? Wait, the user's image shows the fourth option as \(3(n - 3)(n - 2)\)? Wait, let's re - check the factoring.

Wait, original expression: \(3n^2-15n + 18\). Let's factor step by step again.

GCF is 3: \(3(n^2 - 5n+6)\). Now, factor \(n^2 - 5n + 6\). Looking for two numbers that multiply to 6 and add to -5. The numbers are -3 and -2. So, \(n^2-5n + 6=(n - 3)(n - 2)\). So the factored form is \(3(n - 3)(n - 2)\), which is the fourth option. Wait, but let's check the option's writing. The fourth option is \(3(n - 3)(n - 2)\)? Wait, in the image, the fourth option is \(3(n - 3)(n - 2)\)? Let's confirm by expanding \(3(n - 3)(n - 2)\):

First, \((n - 3)(n - 2)=n^2-2n-3n + 6=n^2-5n + 6\). Then multiply by 3: \(3n^2-15n + 18\), which matches the original expression.

Wait, but let's check the other options. Let's expand \(3(n + 5)(n + 2)=3(n^2+7n + 10)=3n^2+21n + 30\) – no. \(3(n + 1)(n + 2)=3(n^2+3n + 2)=3n^2+9n + 6\) – no. \(5(n + 9)(n + 1)=5(n^2+10n + 9)=5n^2+50n + 45\) – no. \(6(n - 10)(n + 2)=6(n^2-8n - 20)=6n^2-48n - 120\) – no. So the correct factored form is \(3(n - 3)(n - 2)\), which is the fourth option.

Answer:

\(3(n - 3)(n - 2)\) (the fourth option in the given list of options)