Solve (5x+2)^2-2x>(5x-4)(5x+4)

$$( 5 x + 2 ) ^ { 2 } - 2 x > ( 5 x - 4 ) ( 5 x + 4 )$$

Solve for x

$$x>-\frac{10}{9}$$

Solution Steps

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5x+2\right)^{2}.

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

Combine 20x and -2x to get 18x.

$$25x^{2}+18x+4>\left(5x-4\right)\left(5x+4\right)$$

Consider \left(5x-4\right)\left(5x+4\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 4.

$$25x^{2}+18x+4>\left(5x\right)^{2}-16$$

Expand \left(5x\right)^{2}.

$$25x^{2}+18x+4>5^{2}x^{2}-16$$

Calculate 5 to the power of 2 and get 25.

$$25x^{2}+18x+4>25x^{2}-16$$

Subtract 25x^{2} from both sides.

$$25x^{2}+18x+4-25x^{2}>-16$$

Combine 25x^{2} and -25x^{2} to get 0.

$$18x+4>-16$$

Subtract 4 from both sides.

$$18x>-16-4$$

Subtract 4 from -16 to get -20.

$$18x>-20$$

Divide both sides by 18. Since 18 is positive, the inequality direction remains the same.

$$x>\frac{-20}{18}$$

Reduce the fraction \frac{-20}{18} to lowest terms by extracting and canceling out 2.

$$x>-\frac{10}{9}$$