Skill: Factor trinomials with a non-unit leading coefficient

To factor a trinomial with a leading coefficient other than $1$, use the product $ac$ and the middle coefficient $b$.

For a trinomial

$$ax^2+bx+c,$$

1. Multiply $a\cdot c$.
2. Find two integers whose product is $ac$ and whose sum is $b$.
3. Rewrite the middle term using those two numbers.
4. Group the four terms and factor each group.
5. Check by multiplying the binomials.

Example: Factor

$$6x^2+11x+3.$$

Here, $a=6$, $b=11$, and $c=3$.

First multiply the first and last coefficients:

$$ac=6\cdot 3=18.$$

Now find two numbers that multiply to $18$ and add to $11$. The numbers are $2$ and $9$:

$$2\cdot 9=18 \qquad\text{and}\qquad 2+9=11.$$

Use $2x$ and $9x$ to split the middle term:

$$6x^2+11x+3 = 6x^2+2x+9x+3.$$

Group the terms:

$$(6x^2+2x)+(9x+3).$$

Factor each group:

$$2x(3x+1)+3(3x+1).$$

Both groups contain the factor $(3x+1)$, so factor it out:

$$(2x+3)(3x+1).$$

Therefore,

$$\boxed{6x^2+11x+3=(2x+3)(3x+1)}.$$

Check by multiplying:

$$(2x+3)(3x+1) =6x^2+2x+9x+3 =6x^2+11x+3.$$

The signs matter: the two numbers must have the correct product and the correct sum. Factoring the first and last terms separately without checking the middle coefficient may produce incorrect factors.