Solve 24x^4+54x^3+30x^2=

$$24 x ^ { 4 } + 54 x ^ { 3 } + 30 x ^ { 2 } =$$

Factor

$$6\left(x+1\right)\left(4x+5\right)x^{2}$$

Solution Steps

Factor out 6.

$$6\left(4x^{4}+9x^{3}+5x^{2}\right)$$

Consider 4x^{4}+9x^{3}+5x^{2}. Factor out x^{2}.

$$x^{2}\left(4x^{2}+9x+5\right)$$

Consider 4x^{2}+9x+5. Factor the expression by grouping. First, the expression needs to be rewritten as 4x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=9 ab=4\times 5=20$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 20.

$$1,20 2,10 4,5$$

Calculate the sum for each pair.

$$1+20=21 2+10=12 4+5=9$$

The solution is the pair that gives sum 9.

$$a=4 b=5$$

Rewrite 4x^{2}+9x+5 as \left(4x^{2}+4x\right)+\left(5x+5\right).

$$\left(4x^{2}+4x\right)+\left(5x+5\right)$$

Factor out 4x in the first and 5 in the second group.

$$4x\left(x+1\right)+5\left(x+1\right)$$

Factor out common term x+1 by using distributive property.

$$\left(x+1\right)\left(4x+5\right)$$

Rewrite the complete factored expression.

$$6x^{2}\left(x+1\right)\left(4x+5\right)$$

Evaluate

$$6\left(x+1\right)\left(4x+5\right)x^{2}$$