Question 3

We will use the given algebraic identity to expand the expressions.

Step 1 — Expanding $(p + 3q + 7r)^2$

Let's identify the terms. Here, $\mathbf{a = p}$, $\mathbf{b = 3q}$, and $\mathbf{c = 7r}$. We use the identity $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.

$(p + 3q + 7r)^2 = (p)^2 + (3q)^2 + (7r)^2 + 2(p)(3q) + 2(3q)(7r) + 2(p)(7r)$

$= p^2 + 9q^2 + 49r^2 + 6pq + 42qr + 14pr$

$\boxed{p^2 + 9q^2 + 49r^2 + 6pq + 42qr + 14pr}$

Step 2 — Expanding $(3x - 2y + 4z)^2$

Let's rewrite the expression first. We can write $(3x - 2y + 4z)^2$ as $[3x + (-2y) + 4z]^2$. Now, let's identify the terms. Here, $\mathbf{a = 3x}$, $\mathbf{b = -2y}$, and $\mathbf{c = 4z}$. We use the identity $(a + b + c)^2 = a^2 + b^2 + c^2 + 2ab + 2bc + 2ca$.

$(3x - 2y + 4z)^2 = (3x)^2 + (-2y)^2 + (4z)^2 + 2(3x)(-2y) + 2(-2y)(4z) + 2(3x)(4z)$

$= 9x^2 + 4y^2 + 16z^2 - 12xy - 16yz + 24xz$

$\boxed{9x^2 + 4y^2 + 16z^2 - 12xy - 16yz + 24xz}$

<DIAGRAM: No diagram needed for this algebraic expansion.>

Answer

(i) $p^2 + 9q^2 + 49r^2 + 6pq + 42qr + 14pr$ (ii) $9x^2 + 4y^2 + 16z^2 - 12xy - 16yz + 24xz$