Let's go through each multiplication problem involving binomials, keeping in mind that $i^2 = -1$.

1) $5i(-4 + 2i)$

Distribute $5i$ to both terms inside the parentheses:

$5i(-4 + 2i) = 5i(-4) + 5i(2i)$ $= -20i + 10i^2$

Since $i^2 = -1$, substitute $-1$ for $i^2$:

$= -20i + 10(-1)$ $= -20i - 10$

The standard form is: $-10 - 20i$

2) $3i(8 - 3i)$

Distribute $3i$ to both terms inside the parentheses:

$3i(8 - 3i) = 3i(8) + 3i(-3i)$ $= 24i - 9i^2$

Substitute $i^2 = -1$:

$= 24i - 9(-1)$ $= 24i + 9$

The standard form is: $9 + 24i$

3) $(2 - i)(3 + i)$

Use the distributive property (FOIL: First, Outer, Inner, Last):

$(2 - i)(3 + i) = 2(3) + 2(i) - i(3) - i(i)$ $= 6 + 2i - 3i - i^2$

Simplify:

$= 6 - i - i^2$

Since $i^2 = -1$, substitute:

$= 6 - i - (-1)$ $= 6 - i + 1$ $= 7 - i$

The standard form is: $7 - i$

4) $(4 + 6i)(9 - 2i)$

Use the distributive property (FOIL method):

$(4 + 6i)(9 - 2i) = 4(9) + 4(-2i) + 6i(9) + 6i(-2i)$ $= 36 - 8i + 54i - 12i^2$

Combine like terms:

$= 36 + 46i - 12i^2$

Substitute $i^2 = -1$:

$= 36 + 46i - 12(-1)$ $= 36 + 46i + 12$ $= 48 + 46i$

The standard form is: $48 + 46i$

---

Final answers in standard form:

1. $-10 - 20i$
2. $9 + 24i$
3. $7 - i$
4. $48 + 46i$

---

Would you like further details on any of the steps?

Here are some additional related problems for practice:

1. Multiply $(1 + i)(1 - i)$.
2. Multiply $7i(4 - 5i)$.
3. Simplify $(3 + 2i)(2 - 4i)$.
4. Multiply $(-6 + i)(5 - 3i)$.
5. Expand $(1 + 3i)^2$.

Tip: Always substitute $i^2 = -1$ early to simplify complex number expressions.