If the equation x2+2(λ+1)x+λ2+λ+7=0has only negative roots, then the least value of λ equals_____.

Let f(x)=x2+2(λ+1)x+λ2+λ+7If both roots of f(x) = 0 are negative, then D=b2−4ac=4(λ+1)2−4λ2+λ+7≥0⇒ λ−6≥0                   → ,λ∈[6,∞)                                (1)Sum of roots =−2(λ+1)<0                         ⇒ λ∈(−1,∞)                          (2)and products of roots =λ2+λ+7>0∀λ∈R    (3)∴  From (1),(2),(3), we get λ∈[6,∞)(as (1), (2), (3) must be satisfied simultaneously.)Hence, the least value of λ=6.