$$
2(2k^2+7k+6)
$$

Factor further:

$$
2(2k+3)(k+2)
$$

$$
3(x^2-25)
$$

Factor further using difference of squares:

$$
3(x-5)(x+5)
$$

$$
2xy(x^2+4x+3y)
$$

b. $3(x-5)(x+5)$

c. $2xy(x^2+4x+3y)$

$$
0=(x-3)(x-4)
$$

Zero product property:

$$
x-3=0text{ or } x-4=0
$$

Solve each equation to $x$:

$$
x=3text{ or } x=4
$$

$$
0=(3x-4)(2x-5)
$$

Zero product property:

$$
3x-4=0text{ or } 2x-5=0
$$

Solve each equation to $x$:

$$
3x=4text{ or } 2x=5
$$

$$
x=dfrac{4}{3}text{ or } x=dfrac{5}{2}
$$

$$
0=(x-3)(x+3)
$$

Zero product property:

$$
x-3=0text{ or } x+3=0
$$

Solve each equation to $x$:

$$
x=3text{ or } x=-3
$$

$$
0=(x+6)^2
$$

Zero product property:

$$
x+6=0
$$

Solve each equation to $x$:

$$
x=-6
$$

b. $left(frac{4}{3},0right)$ and $left(frac{5}{2},0right)$

c. $(3,0)$ and $(-3,0)$

d. $(-6,0)$

$$
y=-15
$$

The $x$-intercept has $y=0$:

$$
0=x^2-2x-15
$$

Factorize:

$$
0=(x-5)(x+3)
$$

Zero product property:

$$
x-5=0text{ or }x+3=0
$$

Solve each equation to $x$:

$$
x=5text{ or }x=-3
$$

These are the $x$-intercepts.

$x$-intercepts: $(5,0)$ and $(-3,0)$

$$
y=a(x-1)(x+1)
$$

We also know that $(0,-1)$ is a point on the graph:

$$
-1=a(-1)(1)=-a
$$

Divide both sides of the equation by $-1$:

$$
1=a
$$

Replace $a$ with 1 in the equation:

$$
y=(x-1)(x+1)
$$

$$
y=a(x+2)(x-5)
$$

We also know that $(0,10)$ is a point on the graph:

$$
10=a(2)(-5)=-10a
$$

Divide both sides of the equation by $-10$:

$$
-1=a
$$

Replace $a$ with $-1$ in the equation:

$$
y=-(x+2)(x-5)
$$

b. $y=-(x+2)(x-5)$

Add 21 to both sides of the equation:

$$
x^2-6x+9=21
$$

Factorize:

$$
(x-3)^2=21
$$

Take the square root of both sides of the equation:

$$
x-3=pm sqrt{21}
$$

Add 3 to both sides of the equation:

$$
x=3pm sqrt{21}
$$

Subtract $4x$ from both sides of the equation:

$$
x^2-4x-3=0
$$

Add 7 to both sides of the equation:

$$
x^2-4x+4=7
$$

Factorize:

$$
(x-2)^2=7
$$

Take the square root of both sides of the equation:

$$
x-2=pm sqrt{7}
$$

Add 2 to both sides of the equation:

$$
x=2pm sqrt{7}
$$

b. $x=2pm sqrt{7}$

$$
P(freshman)=dfrac{64}{212}=dfrac{16}{53}approx 0.302=30.2%
$$

$$
P(sophomore)=dfrac{112}{212}=dfrac{28}{53}approx 0.528=52.8%
$$

Determine the probability using the addition rule:

$$
P(freshmantext{ or }sophomore)=P(freshman)+P(sophomore)-P(freshmantext{ and }sophomore)
$$

$$
=dfrac{16}{53}+dfrac{28}{53}-0=dfrac{44}{53}approx 0.83=83%
$$

b. Determine the probability using the addition rule:

$$
P(bandtext{ or }chorus)=P(band)+P(chorus)-P(bandtext{ and }chorus)
$$

Replace the known probabilities with their values:

$$
75%=dfrac{114}{212}+dfrac{56}{212}-P(bandtext{ and }chorus)
$$

Add $P(bandtext{ and }chorus)$ to both sides of the equation:

$$
P(bandtext{ and }chorus)+dfrac{3}{4}=dfrac{170}{212}
$$

Subtract $dfrac{3}{4}$ from both sides of the equation:

$$
P(bandtext{ and }chorus)=dfrac{170}{212}-dfrac{3}{4}=dfrac{11}{212}approx 0.052=5.2%
$$

$$
x=tan^{-1}dfrac{5}{8}approx 32text{textdegree}
$$

$$
theta = cos^{-1}dfrac{10}{11}approx 24.6text{textdegree}
$$

$$
theta = sin^{-1}dfrac{4}{5}approx 53text{textdegree}
$$

$$
theta = cos^{-1}dfrac{9}{10}approx 25.8text{textdegree}
$$

b. 24.6$text{textdegree}$

c. 53$text{textdegree}$

d. 25.8$text{textdegree}$

$$
sin{40text{textdegree}}=dfrac{h}{600}
$$

Multiply both sides of the equation by 600:

$$
386approx 600sin{40text{textdegree}}=h
$$

$$
triangle ABCsim triangle RTS
$$

$$
triangle ABCsim triangle MPK
$$