Solve {l}{4a^2+4a+1-16a^4}{2a^2+2a-1-4a^3}

$$\left. \begin{array} { l } { 4 a ^ { 2 } + 4 a + 1 - 16 a ^ { 4 } } \\ { 2 a ^ { 2 } + 2 a - 1 - 4 a ^ { 3 } } \end{array} \right.$$

Least Common Multiple

$$\left(2a-1\right)\left(2a^{2}-1\right)\left(16a^{4}-\left(-2a-1\right)^{2}\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$-16a^{4}+\left(2a+1\right)^{2}=-4\left(a-\left(-\frac{1}{4}\sqrt{5}+\frac{1}{4}\right)\right)\left(a-\left(\frac{1}{4}\sqrt{5}+\frac{1}{4}\right)\right)\left(4a^{2}+2a+1\right) 2a^{2}+2a-1-4a^{3}=\left(-2a+1\right)\left(2a^{2}-1\right)$$

Identify all the factors and their highest power in all expressions. Multiply the highest powers of these factors to get the least common multiple.

$$4\left(2a-1\right)\left(a-\frac{1-\sqrt{5}}{4}\right)\left(a-\frac{\sqrt{5}+1}{4}\right)\left(2a^{2}-1\right)\left(4a^{2}+2a+1\right)$$

Expand the expression.

$$64a^{7}-32a^{6}-48a^{5}+8a^{4}+12a^{3}+6a^{2}-2a-1$$

Evaluate

$$-16a^{4}+\left(2a+1\right)^{2},\ -4a^{3}+2a^{2}+2a-1$$