Long division is needed for this part.

If I have time later and no one else has already done it, I will write up the answers

You're welcome Titus. Not sure if Eric has already provided a similar/same answer as below as I can't find the previous question, but you can also do part a) this way :

For 5x³ + x² + 5x + 1,

Notice that 5x³ + x² is similar to 5x + 1.

For 5x³ + x² , we can factorise the x² out first to become x²(5x+1)

Then the expression becomes
x²(5x+1) + 5x + 1, which you can then group together with (5x + 1) as the common factor

So 5x³ + x² + 5x + 1
= x²(5x+1) + 5x +
= (5x+1)(x² + 1)

you can put a 1 in front of the 5x +1 if it helps you to visualise better

The final answer for b) is

x - 4 + (1-x)/(x² + 1) - 3/(5x + 1)

Practise long division on your own to factor out the x and the -4. Then for the remainder that you get, apply the partial fraction template for quadratic factors