Given P = 2i − 3j + k, Q = 3i − 4j − 3k, and R = 3P + 2Q. Find |R| correct to 2 decimal places.

$$P = \begin{bmatrix} 2 \\ -3 \\ 1 \end{bmatrix}$$ $$Q = \begin{bmatrix} 3 \\ -4 \\ -3 \end{bmatrix}$$

The magnitude of a vector $\begin{bmatrix} x \\ y \\ z \end{bmatrix}$ is given by $\sqrt{x^2 + y^2 + z^2}$.

$$|R| = \sqrt{12^2 + (-17)^2 + (-3)^2}$$ $$|R| = \sqrt{144 + 289 + 9}$$ $$|R| = \sqrt{442}$$

$$|R| \approx 21.023796...$$

Rounded to 2 decimal places:

$$|R| \approx \boxed{\text{21.02}}$$

We can factor this quadratic expression by finding two terms that multiply to $2y^2 \times (-2x^2) = -4x^2y^2$ and add to $-3xy$. These terms are $-4xy$ and $xy$.

$$2y^2 - 3xy - 2x^2 = 2y^2 - 4xy + xy - 2x^2$$

Factor by grouping:

$$= 2y(y - 2x) + x(y - 2x)$$ $$= (2y + x)(y - 2x)$$

This is a difference of squares, $a^2 - b^2 = (a-b)(a+b)$, where $a = 2y$ and $b = x$.

$$4y^2 - x^2 = (2y - x)(2y + x)$$

$$\frac{(2y + x)(y - 2x)}{(2y - x)(2y + x)}$$

Cancel out the common factor $(2y + x)$:

$$= \frac{y - 2x}{2y - x}$$

The simplified expression is $\boxed{\frac{y - 2x}{2y - x}}$.

$$3^{2x+3} = 28 - 1$$ $$3^{2x+3} = 27$$

We know that $27 = 3^3$.

$$3^{2x+3} = 3^3$$

Since the bases are the same, the exponents must be equal.

$$2x + 3 = 3$$

$$2x = 3 - 3$$ $$2x = 0$$ $$x = \frac{0}{2}$$ $$x = \boxed{0}$$