Solve 90x^4-26x^3-4x^2

$$90 x ^ { 4 } - 26 x ^ { 3 } - 4 x ^ { 2 }$$

Factor

$$2\left(5x-2\right)\left(9x+1\right)x^{2}$$

Solution Steps

Factor out 2.

$$2\left(45x^{4}-13x^{3}-2x^{2}\right)$$

Consider 45x^{4}-13x^{3}-2x^{2}. Factor out x^{2}.

$$x^{2}\left(45x^{2}-13x-2\right)$$

Consider 45x^{2}-13x-2. Factor the expression by grouping. First, the expression needs to be rewritten as 45x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-13 ab=45\left(-2\right)=-90$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -90.

$$1,-90 2,-45 3,-30 5,-18 6,-15 9,-10$$

Calculate the sum for each pair.

$$1-90=-89 2-45=-43 3-30=-27 5-18=-13 6-15=-9 9-10=-1$$

The solution is the pair that gives sum -13.

$$a=-18 b=5$$

Rewrite 45x^{2}-13x-2 as \left(45x^{2}-18x\right)+\left(5x-2\right).

$$\left(45x^{2}-18x\right)+\left(5x-2\right)$$

Factor out 9x in 45x^{2}-18x.

$$9x\left(5x-2\right)+5x-2$$

Factor out common term 5x-2 by using distributive property.

$$\left(5x-2\right)\left(9x+1\right)$$

Rewrite the complete factored expression.

$$2x^{2}\left(5x-2\right)\left(9x+1\right)$$

Evaluate

$$2\left(5x-2\right)\left(9x+1\right)x^{2}$$