Factorise:

\(x^2+4x+3=(x+1)(x+3)\).

So the roots are \(x=-3\) and \(x=-1\).

Because the coefficient of \(x^2\) is positive, the parabola opens upwards.

An upward-opening parabola is below the x-axis between the roots and above the x-axis outside them.

So the solution is \(-3 < x < -1\).

Factorise:

\(x^2-5x+6=(x-2)(x-3)\).

So the roots are \(x=2\) and \(x=3\).

Because the coefficient of \(x^2\) is positive, the parabola opens upwards.

An upward-opening parabola is below the x-axis between the roots and above the x-axis outside them.

So the solution is \(x \le 2\) or \(x \ge 3\).

Factorise:

\(x^2+x-6=(x+3)(x-2)\).

So the roots are \(x=-3\) and \(x=2\).

Because the coefficient of \(x^2\) is positive, the parabola opens upwards.

An upward-opening parabola is below the x-axis between the roots and above the x-axis outside them.

So the solution is \(x < -3\) or \(x > 2\).

Factorise:

\(x^2-7x+10=(x-2)(x-5)\).

So the roots are \(x=2\) and \(x=5\).

Because the coefficient of \(x^2\) is positive, the parabola opens upwards.

An upward-opening parabola is below the x-axis between the roots and above the x-axis outside them.

So the solution is \(2 \le x \le 5\).

Factorise:

\(2x^2-8x+6=2(x-1)(x-3)\).

So the roots are \(x=1\) and \(x=3\).

Because the coefficient of \(x^2\) is positive, the parabola opens upwards.

An upward-opening parabola is below the x-axis between the roots and above the x-axis outside them.

So the solution is \(1 < x < 3\).