Solve 175x^4+175x^3+28x^2

$$175 x ^ { 4 } + 175 x ^ { 3 } + 28 x ^ { 2 }$$

Factor

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

Solution Steps

Factor out 7.

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

Consider 25x^{4}+25x^{3}+4x^{2}. Factor out x^{2}.

$$x^{2}\left(25x^{2}+25x+4\right)$$

Consider 25x^{2}+25x+4. Factor the expression by grouping. First, the expression needs to be rewritten as 25x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=25 ab=25\times 4=100$$

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 100.

$$1,100 2,50 4,25 5,20 10,10$$

Calculate the sum for each pair.

$$1+100=101 2+50=52 4+25=29 5+20=25 10+10=20$$

The solution is the pair that gives sum 25.

$$a=5 b=20$$

Rewrite 25x^{2}+25x+4 as \left(25x^{2}+5x\right)+\left(20x+4\right).

$$\left(25x^{2}+5x\right)+\left(20x+4\right)$$

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

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

Factor out common term 5x+1 by using distributive property.

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

Rewrite the complete factored expression.

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

Evaluate

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