Solve {l}{(4a^{3}+5)^{2}+(4a^{3}-1)^{2}-2(4a^{3}+5)(4a^{3}-1)}{(p-2a)(p+2a)-(p-a)(p^2+pa+a^2)}{x(2x-1)^2-2(x+1)(x^2-x+1)}

$$\left. \begin{array} { l } { ( 4 a ^ { 3 } + 5 ) ^ { 2 } + ( 4 a ^ { 3 } - 1 ) ^ { 2 } - 2 ( 4 a ^ { 3 } + 5 ) ( 4 a ^ { 3 } - 1 ) } \\ { ( p - 2 a ) ( p + 2 a ) - ( p - a ) ( p ^ { 2 } + p a + a ^ { 2 } ) } \\ { x ( 2 x - 1 ) ^ { 2 } - 2 ( x + 1 ) ( x ^ { 2 } - x + 1 ) } \end{array} \right.$$

Least Common Multiple

$$36\left(x-2\right)\left(2x^{2}+1\right)\left(p^{3}-a^{3}+4a^{2}-p^{2}\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$36=2^{2}\times 3^{2} -2-4x^{2}+x+2x^{3}=\left(x-2\right)\left(2x^{2}+1\right)$$

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

$$36\left(x-2\right)\left(2x^{2}+1\right)\left(p^{3}-a^{3}+4a^{2}-p^{2}\right)$$

Expand the expression.

$$72p^{3}x^{3}-72a^{3}x^{3}+288a^{2}x^{3}-72p^{2}x^{3}+144x^{2}a^{3}-144x^{2}p^{3}+144p^{2}x^{2}-576a^{2}x^{2}+36xp^{3}-36xa^{3}+144xa^{2}-36xp^{2}-72p^{3}+72a^{3}-288a^{2}+72p^{2}$$

Evaluate

$$36,\ a^{3}-p^{3}+p^{2}-4a^{2},\ \left(x-2\right)\left(2x^{2}+1\right)$$