Find the dimensions of a rectangular wooden board with area x2 + 8x + 15 and a table mat with area 216 cm2 where the width is 6 cm less than the length.

We need two numbers that multiply to 15 and add up to 8. These numbers are 3 and 5.

The expressions for the sides of the board are $x+3$ and $x+5$.

The expressions for the sides of the board are $\boxed{(x+3) \text{ and } (x+5)}$.

5. Determine the width of the mat.
The area of a rectangular table mat is $216 \text{ cm}^2$. The width of the mat is $6 \text{ cm}$ shorter than the length.

Let the length of the mat be $L$ and the width be $W$.

Area $A = L \times W = 216$
Width is $6 \text{ cm}$ shorter than the length: $W = L - 6$

Here, $a=1$, $b=-6$, $c=-216$.

We have two possible values for $L$:

Since length cannot be negative, we take $L = 18 \text{ cm}$.

The width of the mat is $\boxed{12 \text{ cm}}$.

6. Determine their present ages.
A mother is five times as old as her daughter. Six years ago, the product of their ages was 68.

Let the daughter's present age be $D$.
Let the mother's present age be $M$.

Mother is five times as old as her daughter: $M = 5D$
Ages six years ago:
Daughter's age six years ago: $D - 6$
Mother's age six years ago: $M - 6$
Product of their ages six years ago was 68: $(D - 6)(M - 6) = 68$

Here, $a=5$, $b=-36$, $c=-32$.

We have two possible values for $D$:

Since age cannot be negative, the daughter's present age is $D = 8$ years.

The daughter's present age is 8 years and the mother's present age is 40 years.