Sure, let's work through this step-by-step.
### Part (a): Fill in the gaps
We begin with the given expression:
[tex]\[ 3x^2 - 6x - 24 \][/tex]
First, factor out the greatest common factor (GCF) from each term. Notice that each term has a factor of 3:
[tex]\[ 3x^2 - 6x - 24 = 3(x^2 - 2x - 8) \][/tex]
This expression in the parentheses is what we need to fill in:
[tex]\[ 3(x^2 - 2x - 8) \][/tex]
So, the gaps are:
[tex]\[ 3(x^2 - 2x - 8) \][/tex]
Thus, the completed equation is:
[tex]\[ 3(x^2 - 2x - 8) \][/tex]
### Part (b): Fully factorise
Now we need to factorise the quadratic expression \(x^2 - 2x - 8\). To do this, we look for two numbers that multiply to the constant term (-8) and add up to the coefficient of the linear term (-2).
The numbers that work are -4 and 2, because:
[tex]\[ (-4) \times 2 = -8 \][/tex]
[tex]\[ (-4) + 2 = -2 \][/tex]
So, we rewrite the quadratic expression as:
[tex]\[ x^2 - 2x - 8 = (x - 4)(x + 2) \][/tex]
Finally, we include the 3 we factored out initially:
[tex]\[ 3x^2 - 6x - 24 = 3(x - 4)(x + 2) \][/tex]
Thus, the fully factorised form of the expression is:
[tex]\[ 3(x - 4)(x + 2) \][/tex]
### Summary
a) The filled gaps in the equation are:
[tex]\[ 3(x^2 - 2x - 8) \][/tex]
b) The fully factorised form is:
[tex]\[ 3(x - 4)(x + 2) \][/tex]
This completes the problem, leaving the quadratic expression in the desired form.
