Solve {l}{(5x^{4}+25x^{3}+2x^{2}+x+4)(x^{3}+2x^{2}-3x-4)}{5x^{7}+10x^{6}-15x^3-20x^4+25x^6+50x^3}

$$\left. \begin{array} { l } { ( 5 x ^ { 4 } + 25 x ^ { 3 } + 2 x ^ { 2 } + x + 4 ) ( x ^ { 3 } + 2 x ^ { 2 } - 3 x - 4 ) } \\ { 5 x ^ { 7 } + 10 x ^ { 6 } - 15 x ^ { 3 } - 20 x ^ { 4 } + 25 x ^ { 6 } + 50 x ^ { 3 } } \end{array} \right.$$

Least Common Multiple

$$5\left(x+1\right)x^{3}\left(5x^{10}+65x^{9}+217x^{8}-68x^{7}-767x^{6}+161x^{5}+421x^{4}-779x^{3}-21x^{2}+64x-112\right)$$

Solution Steps

Factor the expressions that are not already factored.

$$\left(5x^{4}+25x^{3}+2x^{2}+x+4\right)\left(-4+x+x^{2}\right)\left(1+x\right)=\left(x+1\right)\left(x-\left(-\frac{1}{2}\sqrt{17}-\frac{1}{2}\right)\right)\left(x-\left(\frac{1}{2}\sqrt{17}-\frac{1}{2}\right)\right)\left(5x^{4}+25x^{3}+2x^{2}+x+4\right)$$

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

$$5\left(x+1\right)x^{3}\left(5x^{10}+65x^{9}+217x^{8}-68x^{7}-767x^{6}+161x^{5}+421x^{4}-779x^{3}-21x^{2}+64x-112\right)$$

Expand the expression.

$$25x^{14}+350x^{13}+1410x^{12}+745x^{11}-4175x^{10}-3030x^{9}+2910x^{8}-1790x^{7}-4000x^{6}+215x^{5}-240x^{4}-560x^{3}$$

Evaluate

$$\left(x+1\right)\left(x^{2}+x-4\right)\left(5x^{4}+25x^{3}+2x^{2}+x+4\right),\ 5x^{3}\left(x^{4}+7x^{3}-4x+7\right)$$