An observer spots a plane flying at a 42 degree angle to his horizontal line of sight. if the plane is flying at an altitude of 15,000 ft, what is the distance (x) from the plane (p) to the observer (o)

This seems like a high school question. I will use trigonometry.

The three points (P, O, and the point on the ground directly below P [call it G]) form a right triangle. The 42-degree angle is opposite PG, the altitude of the plane. Since we are told to find x, which is PO, we can use the definition of the sine function, which is opposite divided by hypotenuse.

Solving for x,

This makes sense because the hypotenuse must be longer than either of the legs.

the quotient is 4x^2 - 27x + 167 with a remainder of -996

step-by-step explanation:

in synthetic division, use -6 as the divisor and use the coefficients {4, -3, 5, 6}:

-6     )   4     -3     5     6

                -24   162 -1002

          4   -27   167   -996

thus, the quotient is 4x^2 - 27x + 167 with a remainder of -996.

note:   your "4x°-3x+5x+6" is ambiguous / incomplete.   i had to assume that you meant '4x^3 - 3x^2 + 5x + 6."