Ive got some math questions

1. a cubic polynomial f(x) = x^3 +3x ^3 + ax + b, where a and b are constants, has a factor x+ 1. The remainder when f(x) is divided by x + 2 is the same as the remainder when f (x) is divided by x- 2. find this remainder.

2. two planes have equations x + 2y - z = 3 and 2x - z = 0. find
the acute angle between the planes
the coordinates of the two points on the line of intersection, l , of the planes
the equation of l, giving your answer in the form r + a + b.

3. show that integral -1 to 0 of 2x + 1 / ( x - 2)( x^ 2 + 1) dx = -3/ 2 ln 2

4. integral of sec^2 (x) / tan x

5. the angle theta satisfies the equation tan2 theta = sin theta
show that either sin theta = 0 or 2 cos theta = 2 cos ^2 - 1

6. the parametric equations of a curve are
x = t ^3 - e ^ -t, y = t ^ 2 - e ^ -2t^2

find the equation of the tangent to the curve at the point where t = 0

the curve cuts the y axis at the point A
show that the value of t at A lies between 0 and 1

I KNOW THIS IS QUITE A BIT BUT ANY HELP OFFERED WOULD BE GREATLY APPRECIATED

1. a cubic polynomial f(x) = x^3 +3x ^2 + ax + b, where a and b are constants, has a factor x+ 1. The remainder when f(x) is divided by x + 2 is the same as the remainder when f (x) is divided by x- 2. find this remainder.
f(x) = x³ + 3x² + ax +b given perhaps you mad a mistake in copying the problem.
As (x+1) is a factor f(-1) = 0 = -1 +3 -a +b = 0 or a-b = 2 --------------- (1)
Using remainder theorem
f(-2) = f (2) or
-8 + 12 -2a + b = 8 + 12 +2a + b = Required remainder -------------------- (2), or
4a = -16 or a = -4 and (1) gives b = a -2 = -6; substituting in (2)
Required remainder = 20 - 8 - 6 = 6

Others I am solving.

2. two planes have equations x + 2y - z = 3 and 2x - z = 0. find
the acute angle between the planes
the coordinates of the two points on the line of intersection, l , of the planes