Question

It's an important question please help me​

Answer 1

Answer:

Therefore if $P(x^2 + x) +k=0$ has real roots

P = 15

k ∈ (-∞,3.75]

Otherwise  if $P(x^2 + x) +k=0$ has equal roots

P = 15

k = 3.75

Above i have given two answers because in the question you did not state that weather the roots are to be real or they are to be same

Step-by-step explanation:

$f(x)=2x^2 + Px + 15$

since -5 is a root of f(x)

$f(-5) =0\\2(-5)^2 + P(-5) +15=0\\50-5P+15=0\\-5P+75=0\\-5P=-75\\P=\frac{-75}{-5}=\frac{75}{5} = 15$

second equation is

$P(x^2 + x) +k=0\\15x^2 + 15x +k=0$

For real roots

$D \geq 0\\\sqrt{b^2 -4ac} \geq 0\\\sqrt{15^2 -4*15*k} \geq 0\\\sqrt{225-60k} \geq 0\\Therefore \ for \ the \ above \ equation \ to \ be \ valid \\225 \geq 60k\\60k \leq 225\\k \leq \frac{225}{60}\\k \leq 3.75$

Therefore k ∈ (-∞,3.75]

If you don't know what the above statement means here is an explanation it means that  k is lesser than or equal to 3.75 which means

-∞ < k ≤ 3.75