Factorise 75x2 + 10x - 21 and 49p2 - 4q2. Solve 4(2x+4) = 240 and x2 − 16 = 240.

The expression is $75x^2 + 10x - 21$.
We need to find two numbers that multiply to $75 \times (-21) = -1575$ and add up to 10. These numbers are 45 and -35.
Rewrite the middle term:

$$75x^2 + 45x - 35x - 21$$

Group the terms and factor:

$$(75x^2 + 45x) - (35x + 21)$$ $$15x(5x + 3) - 7(5x + 3)$$

Factor out the common binomial $(5x+3)$:

$$(5x + 3)(15x - 7)$$

The factored expression is $\boxed{(5x+3)(15x-7)}$.

The expression is $49p^2 - 4q^2$.
This is a difference of squares, $a^2 - b^2 = (a-b)(a+b)$.
Here, $a = \sqrt{49p^2} = 7p$ and $b = \sqrt{4q^2} = 2q$.

$$(7p)^2 - (2q)^2 = (7p - 2q)(7p + 2q)$$

The factored expression is $\boxed{(7p - 2q)(7p + 2q)}$.

The visible equation is $4(2x+4) = 240$.
Divide both sides by 4:

$$2x+4 = \frac{240}{4}$$ $$2x+4 = 60$$

Subtract 4 from both sides:

$$2x = 60 - 4$$ $$2x = 56$$

Divide by 2:

$$x = \frac{56}{2}$$ $$x = 28$$

Another visible equation is $x^2 - 16 - 240 = 0$.

$$x^2 - 256 = 0$$ $$x^2 = 256$$

Take the square root of both sides:

$$x = \pm\sqrt{256}$$ $$x = \pm 16$$

Assuming $x$ represents a positive dimension, $x = 16$.
The solutions for $x$ from the visible equations are $\boxed{x=28}$ and $\boxed{x=16}$.

The expression for the area is $(3x+1)(2x+5)$.
Expand the product:

$$(3x+1)(2x+5) = 3x(2x) + 3x(5) + 1(2x) + 1(5)$$ $$= 6x^2 + 15x + 2x + 5$$ $$= 6x^2 + 17x + 5$$

The expanded expression is $\boxed{6x^2 + 17x + 5}$.