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Advanced Math / Equivalent expressions Difficulty: Medium

$g (x) = \frac{3}{5} x + \frac{7}{6}$

$h (x) = 6 x - 5$

The functions $g$ and $h$ are defined by the equations shown. Which expression is equivalent to $g (x) \cdot h (x)$ ?

$\frac{18 x^{2}}{5} - \frac{35}{6}$

$\frac{18 x^{2}}{5} + \frac{27 x}{11} - \frac{35}{6}$

$\frac{18 x^{2}}{5} - 4 x - \frac{35}{6}$

$\frac{18 x^{2}}{5} + 4 x - \frac{35}{6}$

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Explanation

Choice D is correct. It’s given that $g (x) = \frac{3}{5} x + \frac{7}{6}$ and $h (x) = 6 x - 5$ . Substituting $\frac{3}{5} x + \frac{7}{6}$ for $g (x)$ and $6 x minus 5$ for $h (x)$ in the expression $g (x) \cdot h (x)$ yields $(\frac{3}{5} x + \frac{7}{6}) (6 x - 5)$ . This expression can be rewritten as $\frac{3}{5} x (6 x - 5) + \frac{7}{6} (6 x - 5)$ , or $\frac{18 x^{2}}{5} - 3 x + 7 x - \frac{35}{6}$ , which is equivalent to $\frac{18 x^{2}}{5} + 4 x - \frac{35}{6}$ .

Choice A is incorrect. This expression is equivalent to $\frac{3}{5} x (6 x) + \frac{7}{6} (- 5)$ , not $(\frac{3}{5} x + \frac{7}{6}) (6 x - 5)$ .

Choice B is incorrect and may result from conceptual or calculation errors.

Choice C is incorrect. This expression is equivalent to $(\frac{3}{5} x - \frac{7}{6}) (6 x + 5)$ , not $(\frac{3}{5} x + \frac{7}{6}) (6 x - 5)$ .