Simplify (2x^2 + 7x + 3)/(x + 3).

Step-by-step solution

1. 1

   Note the restriction before simplifying. The denominator vanishes at $x=-3$, so the expression is undefined there. Whatever the simplified form turns out to be, that exclusion must be carried along with it.

2. 2

   Split the middle term. To factor $2x^{2}+7x+3$, look for two numbers multiplying to $a\cdot c=2\times3=6$ and adding to $b=7$. Those are $6$ and $1$:

   $2x^{2}+7x+3=2x^{2}+6x+x+3.$

   The order of the split does not matter; either arrangement groups successfully.

3. 3

   Factor by grouping. Taking $2x$ out of the first pair and $1$ out of the second:

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

   The appearance of $(x+3)$ — exactly the denominator — is the point of the exercise. Expanding back gives $2x^{2}+6x+x+3$ ✓.

4. 4

   Cancel the common factor. Provided $x\ne-3$, the factor $(x+3)$ divides out:

   $\frac{(2x+1)(x+3)}{x+3}=2x+1.$

   Cancelling is only valid because $x+3\ne0$; this is division, not the setting of a factor to zero.

5. 5

   State the answer with its domain. The simplified expression is

   $2x+1,\qquad x\ne-3.$

   The graph is the straight line $y=2x+1$ with a hole at $x=-3$, where the line would otherwise pass through $(-3,-5)$ — a removable discontinuity. Checking at $x=1$: the original gives $\tfrac{2+7+3}{4}=3$ and the simplified form gives $3$ ✓.