a(t)=(t-k)(t-3)(t-6)(t+3)

Solution for a(t)=(t-k)(t-3)(t-6)(t+3) equation:

Simplifying
a(t) = (t + -1k)(t + -3)(t + -6)(t + 3)

Multiply a * t
at = (t + -1k)(t + -3)(t + -6)(t + 3)

Reorder the terms:
at = (-1k + t)(t + -3)(t + -6)(t + 3)

Reorder the terms:
at = (-1k + t)(-3 + t)(t + -6)(t + 3)

Reorder the terms:
at = (-1k + t)(-3 + t)(-6 + t)(t + 3)

Reorder the terms:
at = (-1k + t)(-3 + t)(-6 + t)(3 + t)

Multiply (-1k + t) * (-3 + t)
at = (-1k * (-3 + t) + t(-3 + t))(-6 + t)(3 + t)
at = ((-3 * -1k + t * -1k) + t(-3 + t))(-6 + t)(3 + t)
at = ((3k + -1kt) + t(-3 + t))(-6 + t)(3 + t)
at = (3k + -1kt + (-3 * t + t * t))(-6 + t)(3 + t)
at = (3k + -1kt + (-3t + t2))(-6 + t)(3 + t)
at = (3k + -1kt + -3t + t2)(-6 + t)(3 + t)

Multiply (3k + -1kt + -3t + t2) * (-6 + t)
at = (3k * (-6 + t) + -1kt * (-6 + t) + -3t * (-6 + t) + t2(-6 + t))(3 + t)
at = ((-6 * 3k + t * 3k) + -1kt * (-6 + t) + -3t * (-6 + t) + t2(-6 + t))(3 + t)
at = ((-18k + 3kt) + -1kt * (-6 + t) + -3t * (-6 + t) + t2(-6 + t))(3 + t)
at = (-18k + 3kt + (-6 * -1kt + t * -1kt) + -3t * (-6 + t) + t2(-6 + t))(3 + t)
at = (-18k + 3kt + (6kt + -1kt2) + -3t * (-6 + t) + t2(-6 + t))(3 + t)
at = (-18k + 3kt + 6kt + -1kt2 + (-6 * -3t + t * -3t) + t2(-6 + t))(3 + t)
at = (-18k + 3kt + 6kt + -1kt2 + (18t + -3t2) + t2(-6 + t))(3 + t)
at = (-18k + 3kt + 6kt + -1kt2 + 18t + -3t2 + (-6 * t2 + t * t2))(3 + t)
at = (-18k + 3kt + 6kt + -1kt2 + 18t + -3t2 + (-6t2 + t3))(3 + t)

Combine like terms: 3kt + 6kt = 9kt
at = (-18k + 9kt + -1kt2 + 18t + -3t2 + -6t2 + t3)(3 + t)

Combine like terms: -3t2 + -6t2 = -9t2
at = (-18k + 9kt + -1kt2 + 18t + -9t2 + t3)(3 + t)

Multiply (-18k + 9kt + -1kt2 + 18t + -9t2 + t3) * (3 + t)
at = (-18k * (3 + t) + 9kt * (3 + t) + -1kt2 * (3 + t) + 18t * (3 + t) + -9t2 * (3 + t) + t3(3 + t))
at = ((3 * -18k + t * -18k) + 9kt * (3 + t) + -1kt2 * (3 + t) + 18t * (3 + t) + -9t2 * (3 + t) + t3(3 + t))
at = ((-54k + -18kt) + 9kt * (3 + t) + -1kt2 * (3 + t) + 18t * (3 + t) + -9t2 * (3 + t) + t3(3 + t))
at = (-54k + -18kt + (3 * 9kt + t * 9kt) + -1kt2 * (3 + t) + 18t * (3 + t) + -9t2 * (3 + t) + t3(3 + t))
at = (-54k + -18kt + (27kt + 9kt2) + -1kt2 * (3 + t) + 18t * (3 + t) + -9t2 * (3 + t) + t3(3 + t))
at = (-54k + -18kt + 27kt + 9kt2 + (3 * -1kt2 + t * -1kt2) + 18t * (3 + t) + -9t2 * (3 + t) + t3(3 + t))
at = (-54k + -18kt + 27kt + 9kt2 + (-3kt2 + -1kt3) + 18t * (3 + t) + -9t2 * (3 + t) + t3(3 + t))
at = (-54k + -18kt + 27kt + 9kt2 + -3kt2 + -1kt3 + (3 * 18t + t * 18t) + -9t2 * (3 + t) + t3(3 + t))
at = (-54k + -18kt + 27kt + 9kt2 + -3kt2 + -1kt3 + (54t + 18t2) + -9t2 * (3 + t) + t3(3 + t))
at = (-54k + -18kt + 27kt + 9kt2 + -3kt2 + -1kt3 + 54t + 18t2 + (3 * -9t2 + t * -9t2) + t3(3 + t))
at = (-54k + -18kt + 27kt + 9kt2 + -3kt2 + -1kt3 + 54t + 18t2 + (-27t2 + -9t3) + t3(3 + t))
at = (-54k + -18kt + 27kt + 9kt2 + -3kt2 + -1kt3 + 54t + 18t2 + -27t2 + -9t3 + (3 * t3 + t * t3))
at = (-54k + -18kt + 27kt + 9kt2 + -3kt2 + -1kt3 + 54t + 18t2 + -27t2 + -9t3 + (3t3 + t4))

Combine like terms: -18kt + 27kt = 9kt
at = (-54k + 9kt + 9kt2 + -3kt2 + -1kt3 + 54t + 18t2 + -27t2 + -9t3 + 3t3 + t4)

Combine like terms: 9kt2 + -3kt2 = 6kt2
at = (-54k + 9kt + 6kt2 + -1kt3 + 54t + 18t2 + -27t2 + -9t3 + 3t3 + t4)

Combine like terms: 18t2 + -27t2 = -9t2
at = (-54k + 9kt + 6kt2 + -1kt3 + 54t + -9t2 + -9t3 + 3t3 + t4)

Combine like terms: -9t3 + 3t3 = -6t3
at = (-54k + 9kt + 6kt2 + -1kt3 + 54t + -9t2 + -6t3 + t4)

Solving
at = -54k + 9kt + 6kt2 + -1kt3 + 54t + -9t2 + -6t3 + t4

Solving for variable 'a'.

Move all terms containing a to the left, all other terms to the right.

Divide each side by 't'.
a = -54kt-1 + 9k + 6kt + -1kt2 + 54 + -9t + -6t2 + t3

Simplifying
a = -54kt-1 + 9k + 6kt + -1kt2 + 54 + -9t + -6t2 + t3

Reorder the terms:
a = 54 + 9k + -54kt-1 + 6kt + -1kt2 + -9t + -6t2 + t3