### Video Transcript

Show that negative four 𝑥 minus 12 over 𝑥 minus four divided by 𝑥 squared plus
seven 𝑥 plus 12 over 𝑥 cubed minus 16𝑥 simplifies to 𝑎𝑥, where 𝑎 is an
integer.

Our first step here is to consider what happens when we divide by a fraction. Well, 𝑎 over 𝑏 divided by 𝑐 over 𝑑 can be rewritten as 𝑎 over 𝑏 multiplied by
𝑑 over 𝑐. We can multiply by the reciprocal of the second fraction. In this case, our question can be rewritten as negative four 𝑥 minus 12 over 𝑥
minus four multiplied by 𝑥 cubed minus 16𝑥 over 𝑥 squared plus seven 𝑥 plus
12.

We now need to simplify the numerators and denominators by factorizing. Negative four 𝑥 minus 12 can be simplified to negative four multiplied by 𝑥 plus
three by factorizing out negative four. This is because negative four multiplied by 𝑥 is negative four 𝑥 and negative four
multiplied by three is negative 12. The numerator of the second fraction 𝑥 cubed minus 16𝑥 has 𝑥 as a common
factor. Therefore, it can be rewritten as 𝑥 multiplied by 𝑥 squared minus 16. This bracket in turn can be factorized again using the difference of two squares. This gives us 𝑥 multiplied by 𝑥 plus four multiplied by 𝑥 minus four.

Finally, we can factorize the denominator of the second fraction into two
brackets. The first term of these brackets will be 𝑥. And the second terms need to have a product of 12 and a sum of seven. Three multiplied by four is equal to 12 and three plus four equals seven. Therefore, our two brackets are 𝑥 plus three and 𝑥 plus four. We can therefore rewrite our expression as negative four multiplied by 𝑥 plus three
over 𝑥 minus four multiplied by 𝑥 multiplied by 𝑥 plus four multiplied by 𝑥
minus four over 𝑥 plus three multiplied by 𝑥 plus four.

Our next step is to try to cancel brackets that appear on the top and the bottom of
the fractions. Firstly, we can cancel 𝑥 plus four in the second fraction. We can also cancel an 𝑥 plus three from the top of the first fraction and the bottom
of the second fraction. Finally, we can cancel 𝑥 minus four from the top of the second fraction and the
bottom of the first fraction. This leaves us with minus four multiplied by 𝑥, which we can simplify to negative
four 𝑥.

Our expression has therefore been simplified to the form 𝑎𝑥, where 𝑎 is an
integer, in this case negative four.