This is a multiple-choice question with one or more correct answers. In this case we must evaluate each of the statements in the answer choices to select the ones that are definitely true.
(A) The line given by the equation $\(8 x+2 y=\frac{2}{3}\)$ forms a right angle with line $L$.

If two lines form a right angle with one another, they will be perpendicular, and the slopes of the two lines are negative inverses of each other, meaning their product is -1 . For line $L$, the slope is 4 .
What's the slope of the line given in this statement? Let's rewrite the equation in slopeintercept form.

$$
\(\begin{aligned}
8 x+2 y & =\frac{2}{3} \\
2 y & =-8 x+\frac{2}{3} \\
y & =-4 x+\frac{1}{3}
\end{aligned}\)
$$


So the slope is -4 , which is the negative of $L$ 's slope but not the negative inverse (which would be $-\(\frac{1}{4}\)$ ). Thus, this line is not perpendicular to $L$. Not true.
(B) The line given by the equation $\(3+y=4 x\)$ is parallel to line $L$.

Two lines that are parallel share the same slope but have different $y$-intercepts. Let's rewrite the equation for the line given in this statement and put it into slope-intercept form.

$$
\begin{aligned}
3+y & =4 x \\
y & =4 x-3
\end{aligned}
$$


The slope is 4 (same as L ), but the $y$-intercept is -3 (not 3 , as for line $L$ ). Thus, the two lines are parallel, and this statement is true.
(C) Line $L$ passes through exactly two quadrants.

The coordinate plane is divided into four quadrants. Because lines continue infinitely, almost all lines pass through exactly three quadrants. However, lines that are vertical or horizontal pass through only two quadrants. Since line $L$ is neither vertical nor horizontal, and does not pass through the origin, it will pass through three quadrants. This statement is not true.

(D) Line $M$ is given by $y=2 x+b$, where $b$ is positive; $L$ 's $y$-intercept is greater than $M$ 's $x$ intercept.
Given that $L$ and $M$ both have a positive slope and a positive $y$-intercept, we know that they both have a negative $x$-intercept (draw it to see why this is the case).
What is $L$ 's $x$-intercept? Calculate it by letting $y=0$.

$$
\(0=4 x+3, x=-\frac{3}{4}\)
$$


For example, if $b=\frac{1}{4}$, then line $M$ is $\(y=2 x+\frac{1}{4}\)$. If $y=0$, then $\(x=-\frac{1}{8}\)$, which is greater than $\(-\frac{3}{4}\)$.
Thus, although this statement is true for most positive values of $b$, it's not true for all of them. For any $\(b<\frac{3}{4}\)$ it will be false, so we can't say this statement must be true.
Only answer choice (B) is correct.