$\displaystyle \left (x+1 \right )^{0}$ is the correct choice.

$\displaystyle 0 (x+1) = 0$ for all values of $\displaystyle x$, since, by the zero property of multiplication, any number multiplied by 0 yields product 0.

$\displaystyle -x + x = 0$ for all values of $\displaystyle x$ - this is a direct statement of the inverse property of addition.

$\displaystyle 0^{x} = 0$, since 0 raised to any positive power yields a result of 0. 

$\displaystyle \left (x+1 \right )^{0} = 1$, since any nonxero number raised to the power of 0 yields a result of 1.

To solve the proportion, cross multiply the terms.

$\displaystyle 3(2x+3) = 1(2)$

$\displaystyle 6x+9 = 2$

Subtract 9 on both sides.

$\displaystyle 6x+9 -9= 2-9$

$\displaystyle 6x=-7$

Divide by 6 on both sides.

$\displaystyle \frac{6x}{6}=\frac{-7}{6}$

The answer is:  $\displaystyle -\frac{7}{6}$

To solve the proportion, cross multiply.

$\displaystyle 5(6) = 9x$

$\displaystyle 30=9x$

Divide by 9 on both sides.

$\displaystyle \frac{30}{9}=\frac{9x}{9}$

Reduce the fractions.

The answer is:  $\displaystyle \frac{10}{3}$

Cross multiply the fractions.

$\displaystyle 3(2x-2) = 8(8x-2)$

Simplify both sides.

$\displaystyle 6x-6 = 64x-16$

Subtract $\displaystyle 6x$ on both sides.

$\displaystyle 6x-6 -(6x)= 64x-16-(6x)$

$\displaystyle -6 = 58x-16$

Add 16 on both sides.

$\displaystyle -6+16 = 58x-16+16$

$\displaystyle 10= 58x$

Divide by 58 on both sides.

$\displaystyle \frac{10}{58}= \frac{58x}{58}$

Reduce both fractions.

The answer is:  $\displaystyle x = \frac{5}{29}$

Cross multiply both sides.

$\displaystyle 5x = 2(3)$

Simplify and solve for x.

$\displaystyle 5x=6$

$\displaystyle \frac{5x}{5}=\frac{6}{5}$

The answer is:  $\displaystyle \frac{6}{5}$

Cross multiply both sides.

$\displaystyle 3(3x) = 1(5)$

$\displaystyle 9x=5$

Divide by 9 on both sides.

$\displaystyle \frac{9x}{9}=\frac{5}{9}$

The answer is:  $\displaystyle \frac{5}{9}$

Cross multiply the two fractions.

$\displaystyle 3(3x) = 8(4)$

$\displaystyle 9x=32$

Divide by nine on both sides.

$\displaystyle \frac{9x}{9}=\frac{32}{9}$

The answer is:  $\displaystyle \frac{32}{9}$

Cross multiply the two fractions.

$\displaystyle (2x)(3) = (7)(9)$

$\displaystyle 6x = 63$

Divide by six on both sides.

$\displaystyle \frac{6x }{6}= \frac{63}{6}$

The answer is:  $\displaystyle \frac{21}{2}$

The product of a 2 x 2 matrix and a 2 x 1 matrix is a 2 x 1 matrix.

Multiply each row in the first matrix by the column matrix by multiplying elements in corresponding positions, then adding the products, as follows:

$\displaystyle \begin{bmatrix} x&7 \\ y& 3 \end{bmatrix}\begin{bmatrix} 2\\ 7 \end{bmatrix}$

$\displaystyle =\begin{bmatrix} x \cdot 2+ 7 \cdot 7 \\ y \cdot 2 + 3 \cdot 7 \end{bmatrix}$\

$\displaystyle =\begin{bmatrix} 2x + 49 \\ 2 y+ 21 \end{bmatrix}$

The product of a 2 x 2 matrix and a 2 x 1 matrix is a 2 x 1 matrix.

Multiply each row in the first matrix by the column matrix by multiplying elements in corresponding positions, then adding the products, as follows:

$\displaystyle \begin{bmatrix} x& 5\\ -4& 3 \end{bmatrix}\begin{bmatrix} 6\\ y \end{bmatrix}$

$\displaystyle = \begin{bmatrix} x \cdot 6 + 5\cdot y \\ -4\cdot 6 + 3 \cdot y \end{bmatrix}$

$\displaystyle = \begin{bmatrix} 6x+5y \\ 3y-24 \end{bmatrix}$