Solve 49/4-a^2/16+a^4-4a^2+(-4a+1)(a-2)(4a+1)(a+2)/16

$$\frac { 49 } { 4 } - \frac { a ^ { 2 } } { 16 } + a ^ { 4 } - 4 a ^ { 2 } + \frac { ( - 4 a + 1 ) ( a - 2 ) ( 4 a + 1 ) ( a + 2 ) } { 16 }$$

Evaluate

$$12$$

Short Solution Steps

Combine -\frac{a^{2}}{16} and -4a^{2} to get -\frac{65}{16}a^{2}.

$$\frac{49}{4}-\frac{65}{16}a^{2}+a^{4}+\frac{\left(-4a+1\right)\left(a-2\right)\left(4a+1\right)\left(a+2\right)}{16}$$

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of 4 and 16 is 16. Multiply \frac{49}{4} times \frac{4}{4}.

$$\frac{49\times 4}{16}-\frac{65}{16}a^{2}+a^{4}+\frac{\left(-4a+1\right)\left(a-2\right)\left(4a+1\right)\left(a+2\right)}{16}$$

Since \frac{49\times 4}{16} and \frac{\left(-4a+1\right)\left(a-2\right)\left(4a+1\right)\left(a+2\right)}{16} have the same denominator, add them by adding their numerators.

$$\frac{49\times 4+\left(-4a+1\right)\left(a-2\right)\left(4a+1\right)\left(a+2\right)}{16}-\frac{65}{16}a^{2}+a^{4}$$

Do the multiplications in 49\times 4+\left(-4a+1\right)\left(a-2\right)\left(4a+1\right)\left(a+2\right).

$$\frac{196-16a^{4}-4a^{3}+64a^{2}+16a+4a^{3}+a^{2}-16a-4}{16}-\frac{65}{16}a^{2}+a^{4}$$

Combine like terms in 196-16a^{4}-4a^{3}+64a^{2}+16a+4a^{3}+a^{2}-16a-4.

$$\frac{192-16a^{4}+65a^{2}}{16}-\frac{65}{16}a^{2}+a^{4}$$

Divide each term of 192-16a^{4}+65a^{2} by 16 to get 12+\frac{65}{16}a^{2}-a^{4}.

$$12+\frac{65}{16}a^{2}-a^{4}-\frac{65}{16}a^{2}+a^{4}$$

Combine \frac{65}{16}a^{2} and -\frac{65}{16}a^{2} to get 0.

$$12-a^{4}+a^{4}$$

Combine -a^{4} and a^{4} to get 0.

$$12$$

Factor

$$2^{2}\times 3$$

Solution Steps

Factor out \frac{1}{16}.

$$\frac{196-a^{2}+16a^{4}-64a^{2}+\left(-4a+1\right)\left(a-2\right)\left(4a+1\right)\left(a+2\right)}{16}$$

Simplify.

$$12$$