For what value of 'a', does the inequality 9a - a² \(\leq\) 17a + 15 holds

Calculation: 

Given: 9a - a2 \(\leq\) 17a + 15

On rearranging

-a² + 9a \(\leq\) 17a + 15

Shifting the sign

a² - 9a \(≥\) - 17a - 15

a2 - 9a + 17a + 15 \(≥\) 0

a² + 8a + 15 \(≥\) 0

a² + 5a + 3a + 15 \(≥\) 0

a(a+ 5) + 3(a + 5)  \(≥\) 0

(a + 3)(a + 5)  \(≥\) 0

So, -3 and -5 are the roots of the equation.

Now look at the diagram given below,

We see that all the numbers less than -5 and all the numbers greater than -3 will give us positive result. While numbers between -5 and -3 will give us negative results

So, all the above value holds for the above equation.