Question

How do you determine whether x+2 is a factor of the polynomial `5x^2+2x+6`?

Answer 1

You see if `-2` is a solution to the polynomial set to 0.

Explanation:

Let's suppose you have this factorized polynomial:
`(x+2)(2x-10)=0`

A property of this factorized polynomial is that each of it's factors when set to 0 is the root to this polynomial.
For example, when `x=-2`, `(x+2)=0` so `-2` is the solution to this polynomial.
This property arises from the property in which when `0` is multiplied with any number, the product is always going to be `0`.

Now, back to your polynomial. Let's set it to 0 first and solve for `x`.
`5x^2+2x+6=0`

Using the quadratic formula, we get `x=(-1+-1sqrt(29)i)/5`

So, `x+2` is not a factor of this equation. We can also evaluate whether `x=-2` will yield `0`, and it does not.

Answer 2

Use the remainder theorem to find `(x+2)` is not a factor.

Explanation:

RemainderTheorem

Given a polynomial `f(x)` and constant `a`, `(x-a)` is a factor of `f(x)` if and only if `f(a) = 0`

`color(white)()`
In our example:

`f(x) = 5x^2+2x+6`

and:

`f(-2) = 5(color(blue)(-2))^2+2(color(blue)(-2))+6`

`color(white)(f(-2)) = 20-4+6`

`color(white)(f(-2)) = 22`

Since this is not zero, `(x+2) = (x-(-2))` is not a factor of `5x^2+2x+6`