Let f(x) = (3x² + x − 5) / (x² + 4). Find f(3), f(2), f(1), f(a+1), f(sqrt(a)), f(a²), f(a) + f(1).

Substitute $x = (b+a)$ into the function:

$$h(b+a) = (b+a)^2 + 2(b+a) - 4$$

Expand the terms:

$$h(b+a) = (b^2 + 2ab + a^2) + (2b + 2a) - 4$$ $$h(b+a) = a^2 + b^2 + 2ab + 2a + 2b - 4$$

The simplified expression is $\boxed{a^2 + b^2 + 2ab + 2a + 2b - 4}$.

Substitute $x = (a-b)$ into the function:

$$h(a-b) = (a-b)^2 + 2(a-b) - 4$$

Expand the terms:

$$h(a-b) = (a^2 - 2ab + b^2) + (2a - 2b) - 4$$ $$h(a-b) = a^2 + b^2 - 2ab + 2a - 2b - 4$$

The simplified expression is $\boxed{a^2 + b^2 - 2ab + 2a - 2b - 4}$.

First, find $h(1+b)$:

$$h(1+b) = (1+b)^2 + 2(1+b) - 4$$ $$h(1+b) = (1 + 2b + b^2) + (2 + 2b) - 4$$ $$h(1+b) = b^2 + 4b + 3 - 4$$ $$h(1+b) = b^2 + 4b - 1$$

Next, find $h(1)$:

$$h(1) = (1)^2 + 2(1) - 4$$ $$h(1) = 1 + 2 - 4$$ $$h(1) = -1$$

Now, substitute these into the expression:

$$\frac{h(1+b)-h(1)}{b} = \frac{(b^2 + 4b - 1) - (-1)}{b}$$ $$\frac{h(1+b)-h(1)}{b} = \frac{b^2 + 4b - 1 + 1}{b}$$ $$\frac{h(1+b)-h(1)}{b} = \frac{b^2 + 4b}{b}$$

Factor out $b$ from the numerator:

$$\frac{h(1+b)-h(1)}{b} = \frac{b(b+4)}{b}$$

Cancel $b$ (assuming $b \neq 0$):

$$\frac{h(1+b)-h(1)}{b} = b+4$$

The simplified expression is $\boxed{b+4}$.

First, find $h(a+b)$:

$$h(a+b) = (a+b)^2 + 2(a+b) - 4$$ $$h(a+b) = a^2 + 2ab + b^2 + 2a + 2b - 4$$

Next, find $h(a)$:

$$h(a) = a^2 + 2a - 4$$

Now, substitute these into the expression:

$$\frac{h(a+b)-h(a)}{b} = \frac{(a^2 + 2ab + b^2 + 2a + 2b - 4) - (a^2 + 2a - 4)}{b}$$

Distribute the negative sign in the numerator:

$$\frac{h(a+b)-h(a)}{b} = \frac{a^2 + 2ab + b^2 + 2a + 2b - 4 - a^2 - 2a + 4}{b}$$

Combine like terms in the numerator:

$$\frac{h(a+b)-h(a)}{b} = \frac{(a^2 - a^2) + (2a - 2a) + (-4 + 4) + 2ab + b^2 + 2b}{b}$$ $$\frac{h(a+b)-h(a)}{b} = \frac{2ab + b^2 + 2b}{b}$$

Factor out $b$ from the numerator:

$$\frac{h(a+b)-h(a)}{b} = \frac{b(2a + b + 2)}{b}$$

Cancel $b$ (assuming $b \neq 0$):

$$\frac{h(a+b)-h(a)}{b} = 2a + b + 2$$

The simplified expression is $\boxed{2a + b + 2}$.