Solve 4a+4a^2+1/4a-8a^2+a-4a^2+a/4a

$$\frac { 4 a + 4 a ^ { 2 } + 1 } { 4 a - 8 a ^ { 2 } } + a - \frac { 4 a ^ { 2 } + a } { 4 a }$$

Evaluate

$$\frac{6a^{2}+3a+1}{4a\left(1-2a\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{4a^{2}+a}{4a}.

$$\frac{4a+4a^{2}+1}{4a-8a^{2}}+a-\frac{a\left(4a+1\right)}{4a}$$

Cancel out a in both numerator and denominator.

$$\frac{4a+4a^{2}+1}{4a-8a^{2}}+a-\frac{4a+1}{4}$$

Factor 4a-8a^{2}.

$$\frac{4a+4a^{2}+1}{4a\left(-2a+1\right)}+a-\frac{4a+1}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{4a\left(-2a+1\right)}{4a\left(-2a+1\right)}.

$$\frac{4a+4a^{2}+1}{4a\left(-2a+1\right)}+\frac{a\times 4a\left(-2a+1\right)}{4a\left(-2a+1\right)}-\frac{4a+1}{4}$$

Since \frac{4a+4a^{2}+1}{4a\left(-2a+1\right)} and \frac{a\times 4a\left(-2a+1\right)}{4a\left(-2a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{4a+4a^{2}+1+a\times 4a\left(-2a+1\right)}{4a\left(-2a+1\right)}-\frac{4a+1}{4}$$

Do the multiplications in 4a+4a^{2}+1+a\times 4a\left(-2a+1\right).

$$\frac{4a+4a^{2}+1-8a^{3}+4a^{2}}{4a\left(-2a+1\right)}-\frac{4a+1}{4}$$

Combine like terms in 4a+4a^{2}+1-8a^{3}+4a^{2}.

$$\frac{4a+8a^{2}+1-8a^{3}}{4a\left(-2a+1\right)}-\frac{4a+1}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4a\left(-2a+1\right) and 4 is 4a\left(-2a+1\right). Multiply \frac{4a+1}{4} times \frac{a\left(-2a+1\right)}{a\left(-2a+1\right)}.

$$\frac{4a+8a^{2}+1-8a^{3}}{4a\left(-2a+1\right)}-\frac{\left(4a+1\right)a\left(-2a+1\right)}{4a\left(-2a+1\right)}$$

Since \frac{4a+8a^{2}+1-8a^{3}}{4a\left(-2a+1\right)} and \frac{\left(4a+1\right)a\left(-2a+1\right)}{4a\left(-2a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4a+8a^{2}+1-8a^{3}-\left(4a+1\right)a\left(-2a+1\right)}{4a\left(-2a+1\right)}$$

Do the multiplications in 4a+8a^{2}+1-8a^{3}-\left(4a+1\right)a\left(-2a+1\right).

$$\frac{4a+8a^{2}+1-8a^{3}+8a^{3}-4a^{2}+2a^{2}-a}{4a\left(-2a+1\right)}$$

Combine like terms in 4a+8a^{2}+1-8a^{3}+8a^{3}-4a^{2}+2a^{2}-a.

$$\frac{3a+6a^{2}+1}{4a\left(-2a+1\right)}$$

Expand 4a\left(-2a+1\right).

$$\frac{3a+6a^{2}+1}{-8a^{2}+4a}$$

Expand

$$-\frac{6a^{2}+3a+1}{4\left(2a^{2}-a\right)}$$

Short Solution Steps

Factor the expressions that are not already factored in \frac{4a^{2}+a}{4a}.

$$\frac{4a+4a^{2}+1}{4a-8a^{2}}+a-\frac{a\left(4a+1\right)}{4a}$$

Cancel out a in both numerator and denominator.

$$\frac{4a+4a^{2}+1}{4a-8a^{2}}+a-\frac{4a+1}{4}$$

Factor 4a-8a^{2}.

$$\frac{4a+4a^{2}+1}{4a\left(-2a+1\right)}+a-\frac{4a+1}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Multiply a times \frac{4a\left(-2a+1\right)}{4a\left(-2a+1\right)}.

$$\frac{4a+4a^{2}+1}{4a\left(-2a+1\right)}+\frac{a\times 4a\left(-2a+1\right)}{4a\left(-2a+1\right)}-\frac{4a+1}{4}$$

Since \frac{4a+4a^{2}+1}{4a\left(-2a+1\right)} and \frac{a\times 4a\left(-2a+1\right)}{4a\left(-2a+1\right)} have the same denominator, add them by adding their numerators.

$$\frac{4a+4a^{2}+1+a\times 4a\left(-2a+1\right)}{4a\left(-2a+1\right)}-\frac{4a+1}{4}$$

Do the multiplications in 4a+4a^{2}+1+a\times 4a\left(-2a+1\right).

$$\frac{4a+4a^{2}+1-8a^{3}+4a^{2}}{4a\left(-2a+1\right)}-\frac{4a+1}{4}$$

Combine like terms in 4a+4a^{2}+1-8a^{3}+4a^{2}.

$$\frac{4a+8a^{2}+1-8a^{3}}{4a\left(-2a+1\right)}-\frac{4a+1}{4}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4a\left(-2a+1\right) and 4 is 4a\left(-2a+1\right). Multiply \frac{4a+1}{4} times \frac{a\left(-2a+1\right)}{a\left(-2a+1\right)}.

$$\frac{4a+8a^{2}+1-8a^{3}}{4a\left(-2a+1\right)}-\frac{\left(4a+1\right)a\left(-2a+1\right)}{4a\left(-2a+1\right)}$$

Since \frac{4a+8a^{2}+1-8a^{3}}{4a\left(-2a+1\right)} and \frac{\left(4a+1\right)a\left(-2a+1\right)}{4a\left(-2a+1\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{4a+8a^{2}+1-8a^{3}-\left(4a+1\right)a\left(-2a+1\right)}{4a\left(-2a+1\right)}$$

Do the multiplications in 4a+8a^{2}+1-8a^{3}-\left(4a+1\right)a\left(-2a+1\right).

$$\frac{4a+8a^{2}+1-8a^{3}+8a^{3}-4a^{2}+2a^{2}-a}{4a\left(-2a+1\right)}$$

Combine like terms in 4a+8a^{2}+1-8a^{3}+8a^{3}-4a^{2}+2a^{2}-a.

$$\frac{3a+6a^{2}+1}{4a\left(-2a+1\right)}$$

Expand 4a\left(-2a+1\right).

$$\frac{3a+6a^{2}+1}{-8a^{2}+4a}$$