(a) Work out the value of $3p - 4t^2$, when $p = 6$ and $t = 5$ - Junior Cycle Mathematics - Question 5 - 2023

Using the distributive property (FOIL method):

1. First, distribute $2x$:

   • $2x * 4 = 8x$
   • $2x * (-5x) = -10x^2$
   • $2x * x^2 = 2x^3$

2. Next, distribute $-3$:

   • $-3 * 4 = -12$
   • $-3 * (-5x) = 15x$
   • $-3 * x^2 = -3x^2$

Now combine all the terms:

$2x^3 - 10x^2 + 8x + 15x - 12 - 3x^2$

Combine like terms:

• $2x^3$
• $(-10x^2 - 3x^2) = -13x^2$
• $(8x + 15x) = 23x$

So, the final expression is:

$2x^3 - 13x^2 + 23x - 12$

First, rearrange the terms for clarity:

$10de + 2d^2 - df - 5ef$

Now, factor by grouping:

1. From the first two terms, factor out $2d$:

   • $2d(5e + d)$

2. From the last two terms, factor out $-f$:

   • $-f(5e + d)$

The expression now looks like:

$2d(5e + d) - f(5e + d)$

Now, factor out the common term $(5e + d)$:

$(5e + d)(2d - f)$

Hence, the fully factorised form is:

$(5e + d)(2d - f)$