Solve L(x)=x^4/4+x^3+x^2

$$L ( x ) = \frac { x ^ { 4 } } { 4 } + x ^ { 3 } + x ^ { 2 }$$

Evaluate

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

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Multiply x^{3}+x^{2} times \frac{4}{4}.

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

Since \frac{x^{4}}{4} and \frac{4\left(x^{3}+x^{2}\right)}{4} have the same denominator, add them by adding their numerators.

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

Do the multiplications in x^{4}+4\left(x^{3}+x^{2}\right).

$$\frac{x^{4}+4x^{3}+4x^{2}}{4}$$

Factor

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

Solution Steps

Factor out \frac{1}{4}.

$$\frac{x^{4}+4x^{3}+4x^{2}}{4}$$

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

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

Consider x^{2}+4x+4. Use the perfect square formula, a^{2}+2ab+b^{2}=\left(a+b\right)^{2}, where a=x and b=2.

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

Rewrite the complete factored expression.

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