Nov 13, 2017

$x {\left(x y + 7\right)}^{3}$

$x {z}^{5}$

$y x + x$ or $x + x \left(y + 4\right)$ or $x + 2 x {y}^{3}$

$x + \left(y + 2\right) z$

#### Explanation:

$x {\left(x y + 7\right)}^{3}$

The question asks for a product of $x$ and a factor that depends on $x$. Therefore you know that:

• You need to have multiplication of $x$ and some other factor.
• Your other factor should change as you change $x$.

Hence, $x {\left(x y + 7\right)}^{3}$ is your best bet, as you have multiplication, and your other factor - ${\left(x y + 7\right)}^{3}$ - changes with $x$.

$x {z}^{5}$

Again, you are asked for a product, but this time your factor does not depend on $x$. Therefore, you know:

• You must have multiplication of $x$ and some other factor.
• Your other factor should not change as you change $x$.

So, $x {z}^{5}$ is your best answer. You have multiplication, but $z$ does not change with $x$.

$y x + x$ or $x + x \left(y + 4\right)$ or $x + 2 x {y}^{3}$

Here, you're asked for a sum of two terms: $x$, and some multiple of $x$. So, you know that:

• You have addition of $x$ and some other term.
• Your other term must be $x$ multiplied by some factor (in terms of any other variable).

You have three answers that fulfill this criteria (listed above): all of them have $x$ added to $x$ times some factor (in terms of $y$).

$x + \left(y + 2\right) z$
Here, you're again asked for a sum, but this time your other term must not depend on $x$. Therefore, you know:
• You have addition of $x$ and some other term.
• Your other term must not have $x$ in it.
$x + \left(y + 2\right) z$ is your best choice here. Note that you have only $y$ and $z$ as variables in your 2nd term, and neither of those depend on $x$, making this your best answer.