Answer
$a=\frac{1}{6}$
Work Step by Step
Both tangent lines are parallel if they have the same slope so:
$$f'(a)=f'(a+1)$$
The first derivative of $f$ is:
$$f'(x)=3x^{2}-4x+1$$
$$3a^{2}-4a+1=3(a+1)^{2}-4(a+1)+1$$
$$3a^{2}-4a+1=3a^{2}+6a+3-4a-4+1$$
$$0=6a+3-4$$
$$0=6a-1$$
$$a=\frac{1}{6}$$
