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

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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.