Which of the following is equivalent to the expression $2x^2y+2x+3xy+3$ ?

Summary Submit Skip Question

Typically, when given an expression with four terms, factor using grouping. Find common factors between pairs of terms:

$$2x^2y+2x+3xy+3$$ $$\underbrace{2x^2y+2x}_\text{$2x$ in common} + \underbrace{3xy+3}_\text{$3$ in common}$$

Factoring each pair,

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

We are left with a single pair of terms, which have $xy+1$ in common, which can be factored out.

$$\underbrace{2x(xy+1)+3(xy+1)}_\text{$xy+1$ in common}$$ $$=\boxed{(xy+1) (2x+3)}$$

Though tedious, we can expand the answer choices. We can demonstrate that the answer choice is suitable because it is equivalent when expanded.

$$(xy+1)(2x+3)$$ $$xy\cdot 2x + xy \cdot 3 + 1\cdot 2x + 1\cdot 3$$ $$=2x^2y+3xy+2x+3$$

The other choices are not equivalent when expanded.

One way to answer this question or remove incorrect choices would be to substitute in a value of $x$ and $y$ into the expressions. For example, if we take $x=0$ and $y=0$, the given expression evaluates to:

$$2x^2y+2x+3xy+3$$ $$2(0)^2(0)+2(0)+3(0)(0)+3$$ $$= 3$$

Using the same values of $x$ and $y$, we can quickly eliminate the last two answer choices since they do not evaluate to $3$.

Repeating for different values of $x$ and $y$ allows us to decide which of the first two answer choices is correct.