Determine Equivalent Forms of (14m/n + 21m²): Expression Analysis

To solve this problem, we'll simplify and compare the given expressions. Let's begin by factoring the original expression:

Factor the original expression $\frac{14m}{n} + 21m^2$:

Notice both terms contain a factor of $7m$:

$\frac{14m}{n} + 21m^2 = 7m\left(\frac{2}{n} + 3m\right)$

Now, let's examine each given choice:

• Choice 1: $7m\left(\frac{2}{n} + 3m\right)$ matches the factored version we derived from the original, so they are equivalent.
• Choice 2: Simplifying $7\left(\frac{2}{nm} + 3m^2\right)$:

  $= 7 \cdot \frac{2}{nm} + 7 \cdot 3m^2 = \frac{14}{nm} + 21m^2$; note this is different from the original expression since we require $\frac{14m}{n}$ in the first term.

• Choice 3: Simplicity is tested for $\frac{m}{n}(14 + 3m)$:

  $= \frac{14m}{n} + \frac{3m^2}{n}$; for equivalence, recall we needed complete $\frac{14m}{n} + 21m^2$, not $\frac{3m^2}{n}$

• Choice 4: Examine $7\frac{m}{n}(2 + 3mn)$:

  Bifurcation gives $= \frac{14m}{n} + 21m^2$; multipliers yield the original expression accurately.

Thus, the answer is clearly seen that both choices 1 and 4 are equivalent to the original expression:

The correct choices are 1 and 4.