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Simplifying 4a^{2}b^{3}+ ab^{4}+ -4ab^{4}+ -1b^{5}= 0 Reorder the terms: ab^{4}+ -4ab^{4}+ 4a^{2}b^{3}+ -1b^{5}= 0 Combine like terms: ab^{4}+ -4ab^{4}= -3ab^{4}-3ab^{4}+ 4a^{2}b^{3}+ -1b^{5}= 0 Solving -3ab^{4}+ 4a^{2}b^{3}+ -1b^{5}= 0 Solving for variable 'a'. Factor out the Greatest Common Factor (GCF), 'b^{3}'. b^{3}(-3ab + 4a^{2}+ -1b^{2}) = 0 Factor a trinomial. b^{3}((a + -1b)(4a + b)) = 0## Subproblem 1

Set the factor 'b^{3}' equal to zero and attempt to solve: Simplifying b^{3}= 0 Solving b^{3}= 0 Move all terms containing a to the left, all other terms to the right. Add '-1b^{3}' to each side of the equation. b^{3}+ -1b^{3}= 0 + -1b^{3}Remove the zero: 0 = -1b^{3}Simplifying 0 = -1b^{3}The solution to this equation could not be determined. This subproblem is being ignored because a solution could not be determined.## Subproblem 2

Set the factor '(a + -1b)' equal to zero and attempt to solve: Simplifying a + -1b = 0 Solving a + -1b = 0 Move all terms containing a to the left, all other terms to the right. Add 'b' to each side of the equation. a + -1b + b = 0 + b Combine like terms: -1b + b = 0 a + 0 = 0 + b a = 0 + b Remove the zero: a = b Simplifying a = b## Subproblem 3

Set the factor '(4a + b)' equal to zero and attempt to solve: Simplifying 4a + b = 0 Solving 4a + b = 0 Move all terms containing a to the left, all other terms to the right. Add '-1b' to each side of the equation. 4a + b + -1b = 0 + -1b Combine like terms: b + -1b = 0 4a + 0 = 0 + -1b 4a = 0 + -1b Remove the zero: 4a = -1b Divide each side by '4'. a = -0.25b Simplifying a = -0.25b## Solution

a = {b, -0.25b}

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