Given:
1 and -1 are the zeroes of the polynomial lx^4 + mx^3 + nx^2 + rx + p.

To prove:
lnp = mr = 0

Explanation:
To prove that lnp = mr = 0, we will use the fact that if a is a zero of a polynomial, then (x - a) is a factor of that polynomial.

1. Zeroes of the Polynomial:
Given that 1 and -1 are zeroes of the polynomial, we can write the polynomial as:
lx^4 + mx^3 + nx^2 + rx + p = l(x - 1)(x + 1)(ax + b)

2. Expanding the Polynomial:
Expanding the polynomial using the distributive property, we have:
lx^4 + mx^3 + nx^2 + rx + p = l(x^2 - 1)(ax + b)

3. Coefficients of the Polynomial:
Now, let's equate the coefficients of the corresponding powers of x on both sides of the equation.

3.1. Coefficient of x^4:
The coefficient of x^4 on the left side is l.
The coefficient of x^4 on the right side is 0 (since there is no x^4 term).
Therefore, l = 0.

3.2. Coefficient of x^3:
The coefficient of x^3 on the left side is m.
The coefficient of x^3 on the right side is 0 (since there is no x^3 term).
Therefore, m = 0.

3.3. Coefficient of x^2:
The coefficient of x^2 on the left side is n.
The coefficient of x^2 on the right side is -l (since (x^2 - 1)(ax + b) = -l(x^2 - 1)).
Therefore, n = -l = 0.

3.4. Coefficient of x:
The coefficient of x on the left side is r.
The coefficient of x on the right side is 0 (since there is no x term).
Therefore, r = 0.

3.5. Constant term:
The constant term on the left side is p.
The constant term on the right side is -lb (since (x^2 - 1)(ax + b) = -lb).
Therefore, p = -lb.

4. Conclusion:
From the above equations, we can conclude that lnp = mr = 0, as l, m, n, r are all equal to 0.