**Factorising 3÷2x² + 16x + 10**

To factorise the given expression, we need to find two binomials whose product is the given expression. Let's proceed step by step.

**Step 1: Look for the Greatest Common Factor (GCF)**
In this expression, we don't have a GCF that can be factored out.

**Step 2: Factorise the quadratic term**
The term 2x² can be factored as (2x)(x).

**Step 3: Factorise the constant term**
The constant term 10 can be factored as 2 * 5.

Now, let's rewrite the expression with the factored terms:

3÷(2x² + 16x + 10) = 3÷[2x² + (8x + 8x) + 10]

**Step 4: Grouping**
Now, we group the terms:

= 3÷[2x² + 8x + 8x + 10]

**Step 5: Factor by grouping**
We can factor by grouping by taking common factors from the first two terms and the last two terms separately:

= 3÷[2x(x + 4) + 2(4x + 5)]

**Step 6: Factor out the common factor**
Now, we can factor out the common factor from each group:

= 3÷[2x(x + 4) + 2(4x + 5)]
= 3÷[2x(x + 4) + 2(4x + 5)]

= 3÷2[x(x + 4) + (4x + 5)]

**Step 7: Combine the terms**
Now, we can combine the terms inside the brackets:

= 3÷2[x² + 4x + 4x + 5]

**Step 8: Factorise the quadratic expression**
The quadratic expression can be factored as (x + 1)(x + 5).

= 3÷2[(x + 1)(x + 5)]

**Step 9: Simplify the expression**
Finally, we can simplify the expression:

= 3(x + 1)÷2(x + 5)

Therefore, the factorised form of the given expression 3÷2x² + 16x + 10 is 3(x + 1)÷2(x + 5).